Mr Wooldridge was a senior political affairs and planning officer at the UN, for which he had worked in Geneva and Liberia before moving to Haiti in 2007.
A Downing Street spokesman said: "The prime minister is hugely grateful for the work that Frederick and others were doing in the UN Stabilisation Mission: helping to build a stronger Haiti, and giving people hope where they had none."
He added that Gordon Brown's thoughts were also with the loved ones of UK nationals whose whereabouts were yet to be confirmed.
It is unclear how many Britons remain unaccounted for in Haiti, after the earthquake which killed tens of thousands.
Steven Fisher, the UK's ambassador to Haiti and the Dominican Republic, said the Foreign Office's rapid deployment team was "doing quite well" in their efforts to account for expatriate Britons.
Relatives say hope is fading for missing UN worker Ann Barnes
But he added: "Even getting out and visiting a few sites in the city takes an awful lot of effort."
The Foreign Office said it was beginning to repatriate UK nationals, and had "located and checked on over 30 other Brits, who have confirmed they are safe and well".
Another UN worker, Ann Barnes, 59, of Leigh-on-Sea, Essex, is feared to have been killed in the collapse of the headquarters in the capital.
The UN has lost at least 40 workers, including the head of the local mission, Hedi Annabi, who has been found dead in the rubble.
Ms Barnes's sister, Irene Marquet, said: "She must be still there in the rubble which is an awful thing to know."
British search-and-rescue workers are to begin hunting for survivors in a town near the epicentre of the earthquake five days ago.
A 71-strong British search-and-rescue team, which includes a number of firefighters and two search dogs, have been deployed the region.
The workers were due to travel to Leogane, about 19km (12 miles) west of Port-au-Prince, at 0900 local time (1400 GMT) on Sunday.
BBC world affairs correspondent Mark Doyle has described the scene in the town as "apocalyptic", with destruction on a scale "even more dramatic" than in the capital.
In Port-au-Prince, the British rescuers have pulled several survivors from collapsed buildings, including a two-year-old girl removed from the rubble of a nursery on Friday.
But Wiltshire firefighter Dan Cooke, who is part of the rescue team, warned: "The people are very tough as a nation and the weather conditions help to keep people alive but as the days go on the chances of people surviving fall massively."
On Sunday, the Bishop of St Albans held a service dedicated to victims of the Haiti earthquake.
In the service in New Barnet, Hertfordshire, the Rt Rev Alan Smith asked "parishes, church schools and chaplaincies throughout the diocese to pray fervently and to give generously".
Britons have so far donated £15m to the Disasters and Emergency Committee, an umbrella group of 13 major British-based charities which has broadcast appeals on TV and radio.
The group says a £25 donation buys a kit of household essentials, £50 is enough to feed a family for a fortnight and £100 funds temporary shelter for two families.
The DEC appeal for victims of the earthquake in Haiti
Separately, the government has allocated £2m to deal with "logistical problems" on the ground in Haiti, such as the provision of cars and lorries to distribute aid, and to improve communications links.
A British Airways plane carrying aid has landed in the neighbouring Dominican Republic, while Virgin Atlantic has sent an airliner carrying aid and medical personnel to Miami, with further flights to Orlando and Jamaica due in the coming days.
Among DEC members' initial responses to the disaster, an Action Aid emergency team is delivering clean water, shelter and blankets to 20,000 people.
Oxfam has 10 tonnes of aid arriving over the weekend, including plastic sheeting, chlorine for drinking water as well as body bags.
The Disasters Emergency Committee is co-ordinating an appeal to help the people of Haiti
There are 13 charities involved including the British Red Cross, Islamic Relief and World Vision
Donate via the
or by telephoning 0370 60 60 900
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