Is this doable?

Doable, yes. Wise? That depends on you. The British Museum is huge, if you decide to do it in this timeframe then have a plan in advance and target a specific area. Westminster Abbey is smaller and you can get a feel for it, but like the museum go in with a plan. I've never been to the War Rooms and can't comment.

Two hours in the British Museum would not be enough time for some, including myself.
Westminster Abbey takes about an hour unless you want to read every single sarcophagus and tomb inscription.
I think two hours is enough time for the Churchill War Rooms.
While, in theory, you might be able to juggle the times and work in all three— the real limiting factor will be how long can you stay on your feet in one day? (And will you really be in the mood for museums all day long or will your brain max out after 6 hours)? I would find it tough and I can walk about 10 miles per day when travelling. And remember you’ll be on your feet even longer taking the tube from your hotel to each of these attractions or you’ll be standing in line for entry.

A better plan might be seeing Westminster Abbey, the outside of the Houses of Parliament/Big Ben and tour the Churchill War Rooms on one day. Then. on another day see the British Museum, Trafalgar Square and the National Gallery., and maybe Piccadilly Circus or Covent Garden—if you’re still standing!

London is so spread out that to best use what time you have you really need to hit attractions geographically and see everything you can while in that particular neighborhood.

I did not visit the British Museum until my 4th visit to London--not a huge fan of antiquities and had my fill of Egyptian stuff long ago--and found that by researching what was in each gallery I could focus on the areas that truly interested me, mainly Britain and Europe, especially the Sutton Hoo artifacts. So I spent around 2 hours total there, including a good poke around their excellent gift shop. 2 hours each in Westminster Abbey and the War Rooms seems about right, depending on how long you linger over exhibits. The War Rooms is really 2 areas--you move through the actual rooms, which was absolutely fascinating to me, and then at the end is a museum devoted to Churchill's life--you can read--or not read--as much as you feel up to.

Doable yes, if your in a time crunch. But don't forget that London is filled with lovely parks and squares to stroll, historic pubs to visit, interesting neighborhoods and more.

The War Rooms is really 2 areas

This is true, and I found the part devoted to Churchill's life underwhelming. And I am a huge Churchill fan! But it was mostly just placards you could read about his life and politics, and you'd get as much info out of books at home without wasting your vacation time. You have to consciously avoid it if you want though, because as I remember, it is right in the middle of the War Rooms, and the normal walking route will bring you through it. If it were me, I would devote all that time to the War Rooms themselves. They have an excellent audio guide with a lot of info for every room.

Having done all 3 more than once, I would give 1 1/2 hr each to Westminster Abbey and War Rooms and add another hour to the British Museum.

Thanks everyone!

There is so much to see in The British Museum. You will be cheating yourselves with just 2 hours. The various reconstructed walls, gates, temples and buildings are fascinating. The Elgin Marbles from the Acropolis are hauntingly beautiful. You should not assume they will still be there the next time you visit. Then there is all the Egyptian stuff. If you have never seen any of that you should take the opportunity. The Sutton Hoo Exhibition is very good and ends this year I think. I haven't been to the Churchill War rooms but honestly, if you are not a huge WWII buff I would cut that in favor of one of the best museums in the world.