I made many trips (five?) to the British Museum in September 2022. It tends to be mobbed, but the upper floor is not as awful as the ground floor. Perhaps October 2023 will be a bit better, but I wouldn't count on it since tourist traffic seems higher this year. I was told by a guard at the museum that weekends are busiest, then Fridays, and that if the day is rainy even more people go to the museum. If you show up at opening time, you'll probably end up in at least a 30-minute security line. Once a week (I think on Friday) the museum is open late. In my single experience with that schedule, the museum was less crowded during the late hours, and a fair number of the visitors seemed to head out before 5 PM. However, there will probably be no food available after 3 or 4 PM.
I'm afraid I don't remember the seating situation. There's a restaurant on the upper floor that's nicer, quieter and more expensive than the grab-and-go place downstairs. (There's apparently a pizza places somewhere inside, as well, but I've never seen it.) The upstairs spot serves afternoon tea as well as lunch. It would be prudent to reserve, though you might get lucky.
The British Museum is free/donation requested, so your best option might be to make multiple shorter visits, though the security line-up would cost you extra time. The place is huge, so you can't possibly see the whole thing in one visit.
I went to the Churchill War Rooms in September 2019. That's a sight for which you should get tickets in advance. The line to buy tickets on the spot can be very long, and I wouldn't be surprised if tickets sell out on some days. The war rooms section is very crowded; it's just a warren of small offices. It takes less than an hour to see that section. The Churchill Museum section can be a lot more time-consuming if you're interested in Churchill. Unlike the war rooms section, the museum section has an open plan, so you're not forced to follow an established walking route. However, there's not a lot of room there, either. I don't think there's any extra space in either section where they could have seats. There's a cafe on site. That would be your best bet for a break.
I went to the Imperial, War Museum at least once in 2017, 2018 and 2019. It's pretty popular, but it doesn't suffer from significant crowding in my experience. It offers a mixture of explanatory material, some sizeable pieces of war materiel and usually one or two good temporary exhibitions. I can't say that I remember seats, but I wouldn't be surprised if there are some. This is another free/donstion requested museum that cannot possible be fully absorbed in one day.