I can only speak for
London, and much of my knowledge may be dated.
For the most part, I'd agree with Fisheyes's recommendations - I would add these considerations:
1) I would skip the Tate* and substitute something like the British Museum, National Gallery, or National Portrait Gallery.
2) I think the "Winston Churchill War Room" might be part of the Imperial War Museum group. Excellent.
3) Do a little research on the neighborhoods you'd want to see - what you like and where you'll already be. King's Road in Chelsea used to be a bit of an eccentric scene that would definitely get the kids' attention.
4) Consider visiting St. Paul's or Westminster Abbey. Even if you're not into old churches or cathedrals, there's a lot of history there, with the monarchy and so forth.
5) Get a
London tourbook and ask your kids for their votes/input.
6) I can't imagine wasting time or money on the Eye.
7) A non-DBR source of advice that I would trust is TripAdvisor. Takes some time to read the reviews but IMO it is well worth the investment.
* because w/r/t modern art, I share Tom Stoppard's
opinion.