DD is a history nerd, she would vote for Tower, Museum of London, Cutty Sark, Maritime Museum (never did all of it but liked it enough to have a nice outing).
Buckingham Palace is a bit disappointing, it is not a lot to see and while we liked it I wouldn't go anytime soon again or even bring visitors unless they are really into it.
She was over the transport museum at that age. Natural History can be a hit and miss, she liked it when she was younger, with 7 she found it either too boring or too detailed. Similar the Science Museum, they were better when she was older and had a better understanding.
The V&A is surprisingly good as well, they do backpack tours of several galleries and we spend 2-3 hours each time and it was never boring.
British Museum - I would say for older children unless there is something she can use for a school topic or they have a special exhibition of a period she is interested in. I also find it big, noisy, full with people and not very easy to go around and find things.