The British Museum has always been very crowded (by museum standards) on my recent visits. Waiting time in line was around 30 minutes shortly after morning opening time in September. The first floor--especially the Egyptian exhibition--seems to be the most crowded. The situation tends to be worse on weekends/holidays (local families, I assume) and on rainy days, or so I have been told.
The Tate Modern and the National Gallery are popular but have not been as mobbed for me as the British Museum. In the National Gallery you can expect to run into localized clumps of people around the most famous/popular paintings, like Van Gogh's Sunflowers, but I've never been uncomfortable there because of crowds. That said, I haven't been to the National Gallery since 2019. The Tate Modern feels airier and was less crowded than the National Gallery on my visits between 2017 and 2019.
Purchasing a (timed) ticket for the Churchill War Rooms in advance is a must; the ticket line tends to be very long, and the ticket you will be able to buy when you get to the head of the line will probably be for an hour or more in the future, not for immediate entry. You might also encounter a total sellout. Be aware that if you're interested in the Churchill Museum part of the CWR, you can easily spend over half a day seeing the entire CWR; in that case, don't opt for a late entry time. I went to the CWR in 2019 and don't know what it's like now; the war rooms themselves were tight quarters and very crowded. The Churchill Museum was also quite busy but had a bit more space. If you only want to see the war rooms, not the Churchill Museum, take a look at the high entry fee and consider whether it will be worth it to you. The Imperial War Museum is excellent and very large. It, too, is popular, but it hasn't been as crowded as the CWR on my visits.
You might consider the Courtauld Gallery. It reopened not too long ago after an extended renovation. It's not on as many people's radar as the National Gallery.
Check websites for all the attractions you're interested in to see what they say about being open, or not, on bank holidays.
If you end up casting around for something interesting to do, consider taking one or more London Walks. They have a broad slate of walking tours priced at 15 GBP each. I see more than a dozen tours listed for May 7 and ten for May 8. They encourage prebooking (you don't currently pay in advance), but I was able to simply show up at the meeting point and take two or three walks in September. I guess prebooking might not be a bad idea for your time period, because if 25 or 30 people showed up, I suppose they might decide to restrict the tour to those who had prebooked. (I'm just speculating.) I should note that there may be adjustments to the list of tours offered while you're in London; just check the website from time to time.