OP here, and thank you so much! This is just the kind of advice I'm looking for, so any other thoughts are welcome. Both DH and I are fairly well-traveled but for whatever reason, London is not a city we've spent much time in or know much about. It's been 12+ years since I've been there myself.
Time frame is about a week - 10 days. I'm thinking that we will stay in one place the entire time and try to do as much as we can from there, without trying to tack on additional stops in the UK. Very open to day trips out of London, however, if you or anyone else has ideas to suggest.
I prefer to go the VRBO/ flat route for accommodations, rather than a hotel, to give us more room and comfort. No real budget set -- I'm not looking to spend money unnecessarily, of course, but not trying to go on the cheap either. (I'm too old for that!)