[JavaBrewer]

IMO as long as you are walking distance from a tube station you will be good to go. I met my son in London 1 month ago. We only had 3 nights with a 6 AM departure so we stayed in a hotel at heathrow airport. The tube/train combo will get you downtown effortlessly. I would imagine it will be rather warm and as Guy pointed out many places do not have AC. London is a very walkable city so bring comfortable shoes. Too many places to recommend. I will say if you like food the Borough Market is a must visit. The Ginger Pig pork rolls are amazing and there are a hundred other options from beer, wine, cheese, you name it. Start your day near the Tower of London and you can pretty much walk everything.

Taking the train from London to Edinburgh is actually relaxing journey, ~ 3 hours or so. We spent 4 nights in Edinburgh and stayed in an AirBnB in the old town close to the castle which was very nice. We use AirBnb nearly exclusively in Europe. Just narrow your search criteria for a "super host" run property. After our short stay in London we met the the girls and used AirBnB in Nice, Prague, Regensburg, and lastly Lucerne. Zero complaints however if you are mobility challenged many ABnB properties are in older/historic buildings and few have elevators.

Have fun and feel free to ask for more info if needed.