I have several times stayed in city or town other than London and taken multiple side trips. This works well for me because I'm pretty patient about 90-minute (or occasionally longer) bus or train trips. YMMV.
Cambridge: went to Saffron Walden, Ely and Kings Lynn.
Norwich: went to Bury St. Edmunds; the Cambridge destinations might also have worked.
Oxford: went to Cotswolds via a GoCotswolds van tour from Moreton-in-Marsh; Blenheim Palace is also very close.
Bristol: went to Bath, Wells and Cardiff.
Chester: took multiple trips into Northern Wales.
York: went to Whitby.
Brighton: went to Arundel and Lewes
I'm sure there are additional worthwhile day trips possible from each of the above bases and that there are many other practical bases.
There's not much in London that sells out way ahead of time, but the Churchill War Rooms are an example of a place where you'd be likely to run into a long ticket line if you just showed up. It was a more-than-half-day visit for me, so I recommend getting a ticket for entry no later than mid-morning and having no firm plans for the afternoon.
The Imperial War Museum is free/donation-requested, and I've never run into a delay at the entry point. It took me multiple visits (over 8 hours in total) to see the whole thing, but I didn't limit myself to the WWII material. My tip is to skip the audio guide, which basically just duplicates the posted explanatory material, unless you prefer to listen rather than read.
For someone with time and an interest in WWII, Bletchley Park is a top day trip. You can spend about the whole day there, so getting an early start is important.
If you happen to get as far north as Liverpool (not on a day trip!), the Western Approaches HQ is also worthwhile.
One thing to watch for in London is special exhibitions on topics of interest to you that might be very popular with the general public. Those things can sell out way in advance if they're related to popular-culture topics (the Royals, rock musicians, etc.) or something like impressionist art. One potentially tricky one is the Cartier jewelry exhibition at the Victoria and Albert Museum. It opens in mid-April and as of today all of April and most weekend days in May are sold out, despite the high price (27 GBP on weekdays, 27 GBP on weekends).
As an experienced traveler (which I know you are as well), I cannot imagine going on a packaged tour to London, which has so many really large museums and other sights where I'd want to spend a lot more time than the tour would allow. Honestly, I think you'd regret it. I'd spend my money on specialized experiences instead. Maybe there's a private tour to a place you'd be interested in that's not usually open to the public or is swamped during normal visiting hours; Westminster Abbey has a pricey early-access option, for example. And I learned today in another thread that there's a 90-GBP tour to the East Wing of Buckingham Palace.
For more casual sightseeing I'm a big fan of London Walks' walking tours. Running time is usually about 2 hours and--with very few exceptions--the tours cost just 20 GBP (less for seniors). The Blitz tour is good. I haven't yet taken Westminster at War but plan to do that this year. There are many other tours with an historical focus; others target specific neighborhoods (Hampstead, Little Venice, Marylebone, etc.), taking you down little streets a tourist wouldn't usually run into. The guides are all licensed and engaging. There are usually at least 10 tours offered per day, selected from among the many dozens on the company's roster.