Q:

How severe was the 1943 Bengal Famine?

I read this in a comment to a newspaper article:

By far the largest British empire human loss in the Second World War was the three million Bengalis who perished in a famine in 1943 that Churchill expressly refused to alleviate with food aid, after years of draining India of food and raw materials. "Winston seems content to let India starve while usi8ng it as amilitary base" remarked Alanbrooke, his chief military adviser. Churchill vetoed Us and Australian offers to send food

Is there any evidence to substantiate this?

A:

In 1943, some 3 million indian subjects of the British Raj died due to bengal famine. 
I think the most authentic and rich source for examining and finding evidences against Churchill in this incident is Madhusree Mukerjee's book, 'Churchill's Secret War', which reveals a side of Churchill's largely ignored in the West and considerably tarnishes his heroic sheen. 
Mukerjee delves into official documents and oral accounts of survivors to paint a horrifying portrait of how Churchill, as part of the Western war effort, ordered the diversion of food from starving
Indians to already well-supplied British
soldiers and stockpiles in Britain and
elsewhere in Europe, including Greece
and Yugoslavia. And he did so with a
churlishness that cannot be excused on
grounds of policy: Churchill's only
response to a telegram from the
government in Delhi about people
perishing in the famine was to ask why
Gandhi hadn't died yet.
British imperialism had long justified
itself with the pretense that it was
conducted for the benefit of the
governed. Churchill's conduct in the
summer and fall of 1943 gave the lie to
this myth. "I hate Indians," he told the
Secretary of State for India, Leopold
Amery. "They are a beastly people with a
beastly religion." The famine was their
own fault, he declared at a war-cabinet
meeting, for "breeding like rabbits."
As Mukerjee's accounts demonstrate,
some of India's grain was also exported
to Ceylon (now Sri Lanka) to meet needs
there, even though the island wasn't
experiencing the same hardship;
Australian wheat sailed past Indian cities
(where the bodies of those who had died
of starvation littered the streets) to
depots in the Mediterranean and the
Balkans; and offers of American and
Canadian food aid were turned down.
India was not permitted to use its own
sterling reserves, or indeed its own ships,
to import food. And because the British
government paid inflated prices in the
open market to ensure supplies, grain
became unaffordable for ordinary
Indians. 

A:

The answers @bhau and @coleopterist gave are good and marshal a lot of important evidence, but there are complementary points of view someone ought to mention - so I guess it falls to me to do this.

Madhusree Mukerjee's findings have been disputed by the eminent Indian economist Amartya Sen. I haven't read both books yet but perusal of the wiki entry about Sen and of this review at the NYRB indicates that Mukerjee's contention is that the famine was caused by inadequate supply (for which the British would be very much culpable), whereas Sen

[...]presents data that there was an adequate food supply in Bengal at
  the time, but particular groups of people including rural landless
  labourers and urban service providers like haircutters did not have
  the monetary means to acquire food as its price rose rapidly due to
  factors that include British military acquisition, panic buying,
  hoarding, and price gouging, all connected to the war in the region.
  In Poverty and Famines, Sen revealed that in many cases of famine,
  food supplies were not significantly reduced. In Bengal, for example,
  food production, while down on the previous year, was higher than in
  previous non-famine years. Thus, Sen points to a number of social and
  economic factors, such as declining wages, unemployment, rising food
  prices, and poor food-distribution systems. These issues led to
  starvation among certain groups in society.
  (quote from here)

If Sen's analysis is correct, then - as far as I can tell - the British are guilty of a sin of omission rather than comission. (I have no wish to do apologetics but there is a difference).

Churchill's comments, as quoted before, are crass and certainly do tarnish his great reputation. However, I think they need to be placed in some sort of context as well. Churchill was hell-bent on winning the war, and winning it in Europe first. Therefore, he focused his attention on this, and actually bothered very little with Indian issues (or Australian issues , for that matter - Australia had committed most of its army to the British war effort in the Mediterranean, on the standard imperial assurance that the RN would protect it from the Japanese, only to find out that this did not quite work out as promised). A hugely telling quote is from Leo Amery's (the Secretary of State for India) diary for November 1944:

It is terrible to think that in nearly five years, apart from
  incidental talk about appointments etc he has never once discussed
  either the Indian situation generally or this sterling balance
  question with me, but has indulged in wild and indeed hardly sane
  tirades in Cabinet.

(Quote taken from p.88 in The last thousand days of the British Empire by Peter Clarke).
To me this indicates that (A) Churchill's shamefully cavalier attitude to the famine in Bengal sprung not from a special animus towards the Indians, but rather from his dogged pursuit of a single objective (VE) on the one hand and from his slapdash working habits on the other. (B) His "tirades" on the subject were not really the same thing as actual British policy and must be read more as rhetorical exercises.
All this, of course, should not obscure two simple points:

There was a terrible famine.
The British, as India's rulers at the time, bear some sort of responsibility for this humanitarian disaster.

P.S.
For a nuanced and comprehensive study of Churchill's attitudes to Empire, as they evolved over time, I recommend the book Churchill's Empire by Richard Toye.

A:

tl;dr
Was the loss of life in the Bengal famine of 1943 the largest British empire human loss in the Second World War?
Yes, without doubt.
Did Churchill expressly refuse to alleviate the famine with food aid, or veto US and Australian offers to send food?
Absolutely not. The evidence shows that statement is completely untrue, although it might be argued that he might have done more to alleviate the situation, had he been blessed with the gift of 20/20 hindsight (like his modern critics).
Indeed, as the historian Arthur Herman wrote:

“We might even say that Churchill indirectly broke the Bengal famine by appointing as Viceroy Field Marshal Wavell, who mobilised the military to transport food and aid to the stricken regions (something that hadn’t occurred to anyone, apparently).”

[Quoted in Langworth, 2017, p150]

Origins of the claim
The claim that Churchill was responsible for the 1943 Bengal famine stems from the book Churchill's Secret War, By Madhusree Mukerjee.
The problem is that the evidence doesn't actually support that conclusion. On the contrary, it actually appears that Churchill did everything he could in the midst of a world war to save the Bengalis, and that without his actions the famine might have been worse.
What is more, the surviving documents show that Churchill explicitly requested assistance from Australia and from the United States. 

Background to the Famine
There were undoubtedly a number of factors that came together to cause the 1943 Bengall famine. Many of these are covered in some detail in the Wikipedia article on the subject. It is also extremely difficult, if not impossible, to assign a definitive starting date to the actual onset of the famine. This is particularly true since different districts in Bengal suffered the effects at different times and to varying degrees. The Government of India dated the onset of full-scale famine to May 1943.
However, there is some uncertainty about quite how much was known in London, and when, about the severity of the famine. In his 1990 book, Bengal Tiger and British Lion: An Account of the Bengal Famine of 1943, Richard Stevenson laid a great deal of the blame at the door of the then Viceroy, Victor Hope, 2nd Marquess of Linlithgow.
Certainly, the lack of reliable statistics does appear to have been a significant factor in the government's apparent reluctance to act earlier.

The Response of Churchill and the War Cabinet
What we do know is that in a report to the War Cabinet on 4 August 1943, the Secretary of State for India, Leo Amery, noted the spread of famine in Bengal. In his briefing, he specifically stressed the effect upon Calcutta and the potential effect on the morale of European troops stationed in India. At this stage, the cabinet offered only a relatively small amount of additional food shipments. Indeed, they explicitly referred to it as "a token shipment".
Three weeks later, The Statesman newspaper published graphic images of starving famine victims in Calcutta, bringing the situation to the attention of the world. It was probably several weeks before copies of the newspaper reached London.
Churchill appointed Field Marshal, Lord Wavell as Viceroy and Governor of India on 1 October 1943. In briefing the cabinet on Wavell's appointment, Churchill stated that Wavell's duty was to:

"... make sure that India was a safe base for the great operations against Japan which were now pending, and that the war was pressed to a successful conclusion, and that famine and food difficulties were dealt with.”

[War Cabinet, 7 October 1943, (Cabinet papers, CAB 65/36/4)]

He then wrote to Wavell:

"Peace, order and a high condition of war-time well-being among the masses of the people constitute the essential foundation of the forward thrust against the enemy ... The hard pressures of world-war have for the first time for many years brought conditions of scarcity, verging in some localities into actual famine, upon India. Every effort must be made, even by the diversion of shipping urgently needed for war purposes, to deal with local shortages….Every effort should be made by you to assuage the strife between the Hindus and Moslems and to induce them to work together for the common good."

He stated that the goal was to be:

“the best possible standard of living for the largest number of people.”

[Winston S. Churchill to Members of the War Cabinet, 8 October 1943.
(Churchill papers, CHAR 23/11)]

In terms of famine relief, Churchill initially urged Australia to provide assistance. In response, Australia promised to supply 350,000 tons of wheat.
The Canadian Prime Minister, MacKenzie King, also offered to provide aid, but Churchill replied that:
“Wheat from Canada would take at least two months to reach India whereas it could be carried from Australia in 3 to 4 weeks.”
Winston S. Churchill to William Lyon Mackenzie King, 4 November 1943.

[Prime Minister’s Personal Telegram T.1842/3 (Churchill papers, CHAR 
20/123/52)].

In India, Viceroy Field Marshal Lord Wavell, then mobilised the military to transport food and other aid to the stricken areas. 
When in 1944, the Secretary of State for India, Leo Amery, requested a further one million tons of grain to ease the ongoing famine, Churchill stated that:

“for the four years ending 1941/42 the average consumption was 52,331,000 tons, i.e., 2½ million tons less than the figure cited by the Secretary of State. This difference would, of course, more than make good the 1½ million tons calculated deficit.”

Furthermore, he noted that diverting a further million tons of grain at that time would not be practicable:

“given the effect of its diversion alike on operations and on our imports of food into this country, which could be further reduced only at the cost of much suffering.”

[War Cabinet, 7 February (Cabinet papers, CAB 65/41)].

One piece of evidence that is missing from most of the modern claims that Churchill was responsible for the famine, is the observation made by the War Cabinet report that the shortages in Bengal had been:

“partly political in character, caused by Marwari supporters of Congress [Gandhi’s party] in an effort to embarrass the existing Muslim Government of Bengal.” 

Another cause, they added, was corrupt local officials:

“The Government of India were unduly tender with speculators and hoarders.”

[Ibid]

The speculation mentioned had arisen after the Japanese invasion of Burma in 1942 had cut off India’s main supply of rice imports.

Nonetheless, the records show that Churchill and the War Cabinet continued to do their best to divert available resources to provide assistance to India. Shipping remained one of the key problems, and the cabinet recommended that:

(a) A further diversion to India of the shipments of food grains destined for the Balkan stockpile in the Middle East. This might amount to 50,000 tons, but would need War Cabinet approval, while United States reactions would also have to be ascertained;
(b) There would be advantage if ships carrying military or civil cargo from the United States or Australia to India could also take a quantity of bagged wheat.

[War Cabinet, 21 February 1944 (Cabinet papers, CAB 65/41)].

In April 1944, we know that Wavel was reporting that the situation in India was still dire. At this point, Churchill even wrote to President Roosevelt to ask for assistance:

I am seriously concerned about the food situation in India ... Last year we had a grievous famine in Bengal through which at least 700,000 people died. This year there is a good crop of rice, but we are faced with an acute shortage of wheat, aggravated by unprecedented storms ... By cutting down military shipments and other means, I have been able to arrange for 350,000 tons of wheat to be shipped to India from Australia during the first nine months of 1944. This is the shortest haul. I cannot see how to do more.
I have had much hesitation in asking you to add to the great assistance you are giving us with shipping but a satisfactory situation in India is of such vital importance to the success of our joint plans against the Japanese that I am impelled to ask you to consider a special allocation of ships to carry wheat to India from Australia ... We have the wheat (in Australia) but we lack the ships. I have resisted for some time the Viceroy’s request that I should ask you for your help, but ... I am no longer justified in not asking for your help.

Winston S. Churchill to President Franklin D. Roosevelt, 29 April 1944.

[Prime Minister’s Personal Telegram T.996/4 (Churchill papers,
CHAR 20/163/106-107)].

Roosevelt replied to Churchill saying that while he had his “utmost sympathy”, his Joint Chiefs had said they were:

“... unable on military grounds to consent to the diversion of shipping ... Needless to say, I regret exceedingly the necessity of giving you this unfavorable reply.”

Roosevelt to Churchill, 1 June 1944. 

[Prime Minister’s Personal Telegram T.1176/4 (Churchill papers,
CHAR 20/165/82)].

Of course, it must be remembered that this was in the context of America's war against Japan in the Pacific and the build-up to D-Day in the European theatre.

Sources
Primary

UK National Archives usage guidance for Cabinet Papers.
Churchill collections.

NOTE:
Copies of many of the (now declassified) War Cabinet papers are available for (free) download as scanned pdf files from the UK National Archives (link above).
Summaries of the papers in the Churchill collection are available on the link above. Access to copies of the documents themselves is only available from libraries with a subscription to the collection.
Secondary

Langworth, Richard M: Winston Churchill, Myth and Reality: What He
Actually Did and Said, McFarland, 2017
Stevenson, Richard: Bengal Tiger and British Lion: An Account of the
Bengal Famine of 1943, Lionheart Press, 1990

