I am in the same situation with ages. If you want to be focused on the kids, go to camden lock markets and look for the horse market area. Keep walking around that section. I also like http://www.visitspitalfields.com/ for Thursday Antique market. www.portobelloroad.co.uk/ is obvious. You must have been to this place.
graysantiques.com/ This is a nice place to visit and great area to discover unique London shopping. Make sure you visit the river in basement. Of course, this is my view. You have really great shopping also in Mayfair as well as Brick Lane. Watch buying mass produced items as they are available anywhere these days online.