The British Museum is still free to get into (probably the only thing in London that is).
Actually all the nationally-owned museums are-- The National Gallery, the Victoria and Albert Museum, the Natural History Museum, the Imperial War Museum, the Museum of London, the exhibitions at the British Museum... it gets really hard to pay for museums anywhere else after you've seen so many showstoppers for free!! There's also some wonderfully quirky smaller (free) museums such as the Wellcome Collection (sort of based on the interactions between medicine/medical related stuff and society and/or art, and including items like Napoleon's toothbrush) and the Hunterian Museum (the museum of surgery... lots of weird stuff in jars). I'd reccomend getting a travel book or two (I like Time Out London the best for the "insider" view--almost all of it is also on their website) and reading about the sights from the eyes of your characters, picking out what specifically they would like to see. London seriously has stuff for everyone, and if you just flock to the same tourist sites as everyone else because you feel you should, you're passing up the chance for a more meaningful personal experience. I'd say pick the sights based on your characters-- maybe they are the kinds of people who feel obligated to see the major tourist sights, but it might also be a chance to show off interesting nuances of character. One of my best friends only had 8 or so hours in London her first time, on a layover--the one thing she picked to do was the Victoria and Albert Museum (the museum of decorative arts), because of her strong interest in design. I personally have been to London at least ten times, and only visited the British Museum once, while going to the V & A, the Natural History Museum, and the National Gallery on almost every trip. I've never been on the London Eye. If you can personalize your characters' experience of London, even a little, you can draw even more story meaning from their couple days there.
Also, I don't know if anyone noted the parks.... there are several big, very nice parks in Central London, notably Hyde Park and Regent's Park. Actual Londoners use these parks all the time too, going for runs, walking dogs, walking kids, playing cricket. They are lovely places to just sit and chill in a busy city, and really some of my fondest memories of London are just sitting or walking in the park.
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