In London, Brick Lane in the East End for Indian food. Just go. I don't even like Indian food that much, and it was really cool. Slightly sketchy, so look cosmopolitan and walk with confidence, and don't take anything too valuable. Remember the Tube stops at midnight, so finish up dinner before 11 PM so you can get home (does this sound like the voice of experience talking? Trying to catch a bus home from the East End at midnight was an experience that I really never want to repeat. But the food was worth it.)
Camden Town... great flea market on the weekends. just get on the Tube, and get off at the Camden Town stop on the weekend. It's GREAT. Everyone goes. There was a really nice carribean fusion restaurant there called Mango, (or it might have been the Mango Room) which I've been to twice, and enjoyed both times. One thing to remember-- the Camden Town tube stop is in Zone 2 on the tube, so make sure you get a zone 2 ticket. The po-lice love to ticket out-of-towners riding there on the local zone tube tickets. I got slammed for a 15 pound fine. It was clearly the London version of a speed trap.
Portobello Road Market. You'll have to check for times-- mainly weekends, I think. If you have ever read Paddington Bear, this is that market. Or seen Notting Hill-- that's it too. Antiques, food, clothing... pretty much anything. It's great.
It's probably too late in the season, but there is an open air theater in ... I think it is Regent's Park. They have wonderful open air theater performances there-- was probably the best live theater I have ever seen in my life. The tickets are woefully expensive-- as I recall somewhere in the 70-80 dollar range; however, they have first come first serve "lawn" seating (the lawn essentially being the grassy areas on the sides of the amphitheather) for like 6 pounds.
If you go to the British Museum, beware the gypsy children, who will try and stop you, distract you with some magazine thing, and steal your wallet. Do not stop, even if they get in your path.
Even the touristy stuff is fun in London. Skip the changing of the Guard.
I largely slept during my visit to Edinburgh (prounced Edin-borro) so I can't make many recommendations about that town.
Have a great trip.