Have just got back from a day out in London with our DD's aged 3 and 6. Among other things, we went to the Bank of England Museum, which is around the back of the Bank of England. It's free to go in, which I think is astonishing for central London!
The museum is not very big - basically two main rooms plus corridors and side rooms. We were there for just over an hour, including a 15 minute storytelling session.
They have well-targetted trails/spotting sheets, for different age groups - both girls really enjoyed finding things on their sheets and when they handed back the pencils at the end, they got a nice little "gold bar" badge and a postcard. There's a gold bar you can handle and try and lift. And this afternoon, they had an actor doing story telling sessions every hour, also free. There were one or two hands on displays, but the trails were the main attraction as far as our kids were concerned.
For grown-ups, the architecture is lovely and there's lots of interesting displays about the development of money and the history of the Bank. Not a subject I'm desperately interested in, but I'd have liked to do it at adult pace because it was fascinating stuff.
Only downside is no cafe. Website is here