There are two different kinds of things called a "router", with two different pronunciations, originating from two different verbs "route" and "rout".
The confusing part of this is that the two pronunciations overlap the two different things.
A router as above (computer thing, from verb "route"), or anything else which routes something, is "rooter" or "rowter" depending on how you pronounce "route" (US English has both "root" and "rowt",
British English has only "root") A router (woodworking tool, from verb "rout", an electrical one is shown above)
is "rowter", never "rooter", in both the US and the UK and other countries.
The dictionary entry may be referring to the wood tool only.
Source: Oxford Online Dictionary & Cambridge Dictionary
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The New Oxford American Dictionary (NOAD) reports that router has two different meanings.
router 1 /ˈraʊdər/ noun A power tool with a shaped cutter, used in carpentry for making grooves for joints, decorative moldings, etc.
router 2 /ˈraʊdər/ noun A device that forwards data packets to the appropriate parts of a computer network.
In both the cases, the pronunciation is the same.
As comparison, the pronunciation of route (as reported from the NOAD) is /rut/, /raʊt/; the pronunciation of rout is /raʊt/.
In British English, the word is pronounced /ˈraʊtə/ when it has the first meaning, and /ˈruːtə/
in the second case.
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Where I live both things are called "rowter" even though we say "root" 66, so if they lived in the U.S. That would be standard.
If they said "rooter" it might sound like a thing used to clean drains.
english.stackexchange.com/users/914/Picturepocket
A router (rooter) routes.
A router (rauter) routs.
To my ear, the American pronunciation is really grating.
english.stackexchange.com/users/5130/Concrete Gannet
I was born in the US.
I pronounce the computer apparatus "rooter" as in "hooter".
I just had a manager at work try to correct me today, but I stuck to my guns.
english.stackexchange.com/users/15940/Regina Moultrie
A rooter is used to clean pipes (specifically to remove roots, or other obstructions).
Because that pronunciation exists, router is pronounced as indicated in the other answers.
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The Cambridge Advanced Learner's Dictionary (CALD) does not include the word 'router'
However, the meaning of route has been described as the following:
route UK /rut/
US /rut/ , UK /raʊt/ noun
[ C ] a particular way or direction between places
The route we had planned took us right across Greece.
I live on a bus route so I can easily get to work.
a method of achieving something
A college education is often the best route to a good job.
US ( UK round )
a set of regular visits that you make to a number of places or people, especially in order to take products as part of your job route UK /rut/
US /rut/ , UK /raʊt/ verb [ T usually + adv/prep ] to send
Deliveries are routed via/by way of London.
Since the CALD did not specify from which sound the word 'router' is originated, so my conclusion is 'router' can be pronounced as 'ruter' or 'rauter' according to UK pronunciation!
However, if the company which produced the device pronounces it as a 'rauter' then it is 'rauter' otherwise it is 'ruter'.
english.stackexchange.com/users/18282/Alex
Merriam-Webster lists both  ˈrüt  and  ˈrau̇t  as possible pronunciations for  route  but only ˈrau̇-tər  for  router .   Is it really wrong to pronounce  router  as  'rüter  ?
What is the correct way to pronounce 'router'?
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<pronunciation>
english.stackexchange.com/questions/2389
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There are two different definitions for the word "router" and this may lead to different pronunciations, although the New Oxford American Dictionary provides the same pronunciation for both. In British English it is always "rooter" but in US English the computer device is a "rowter" and the drain cleaning or carpentry tool is a "rooter".
There are two different definitions for the word "router" and this may lead to different pronunciations, although the New Oxford American Dictionary provides the same pronunciation for both. There are two different kinds of things called a "router", with two different pronunciations.  A  router  (computer thing), or anything else which routes something, is "rooter" or "rowter" depending on how you pronounce "route" (US English has both "root" and "rowt",
There are two different kinds of things called a "router", with two different pronunciations, originating from two different verbs -  "route" and "rout". The computer apparatus  is pronounced as in "hooter".  The word "router" can refer to one of two different items and this may have a bearing on the pronunciation of the word. It is possible that both pronunciations are acceptable and the one you choose may be a matter of personal choice.
There are two different things called a "router" and the pronounciation varies in US English but not in British english. a "router" can be a computer device that forwards data or a carpentry tool that makes grooves. In the US both are called "rowter" and a "rooter" is a tool for cleaning drains. The rowter pronounciation sounds wrong to English ears. But some in the US say rooter.
There are two different kinds of things called a "router", with two different pronunciations.  A  router  (computer thing), or anything else which routes something, is "rooter" or "rowter" depending on how you pronounce "route" (US English has both "root" and "rowt",
There are two different definitions for the word "router" which may have a bearing on its pronunciation. The dictionary entry in question may be referring to only one definition of the word. Some people pronounce "router" the same as "hooter". It is possible that either pronunciation is acceptable.