I spent time in both London and Edinburgh a few months ago. It was done on an open jaw reward ticket (on American; I flew to Glasgow and took the train to Edinburgh, leaving the UK from Heathrow). I used trains to travel in between (I also stopped in York and spent an afternoon in Leeds at the Royal Armoury).
I used hotels; many were reward nights, so can't help on the AirBnB front.
Edinburgh was not a reward night, but I booked almost a year in advance (and only a few months after the Brexit vote, so got really good rates - about $75 a night for a 4* hotel) on a nice hotel right on Queen Street (The Place). It basically allowed a walk to many places (I didn't eat there - too expensive). Eden's Kitchen (a short walk) was good Pizza, albeit small and hard to get a seat :)
I did stay in central York (within the city walls) since it was easy to walk everywhere.
In London, I stayed in Wandsworth, very close to Wandsworth Town station. It made for a nice commute into London (easy access to Victoria or Waterloo). It's also one stop from Clapham Junction, where you can connect to just about everything South and West of London (without having to fully go into the city itself).
Wandsworth also provides a "back method" of getting out to Heathrow that didn't require the Piccadilly line [which would have been insane trying to go toward London with luggage during rush hour]... instead, head away from London toward Woking, getting off at Feltham and take the 490 bus to Heathrow.
I had considered staying in Croyden, as there are hotels very near East Croyden station (and it provides easy access to both Victoria and London Bridge), but with train strikes on Southern last fall, I didn't risk it, and converted my hotel to Wandsworth about a month before the trip, even though it was more money. (rumor has it Wandsworth is also nicer - Croyden has a bit more of a reputation from what I understand)
I spent a bunch of time learning the travel pathways in both places and in London, it heavily depends on where you intend to go in the city... and what you expect to spend in travel time anyway. For example, the train from Wembley Stadium to Marylebone station is only 13 minutes, but it might take another 40 to take the tube to Buckingham Palace.... so the commute "in" isn't the expensive part of the travel time you will spend... just for context :)
I'm happy to provide more thoughts on location if you have the list of things you are considering doing, or drop a note and I can send more info of what I did :)