If you're looking for something a little bit different then Art From the Heart might suit you. They have a Harrogate address although they're not in the centre of town - you'll find opening hours, directions, details of workshops etc on their web site:
Art from the Heart. They stock the likes of the UK company Crafty Individuals and also have their own "Blonde Moments" range of papers and paints etc so you'll probably see things you wouldn't get easily in the US as well as seeing the more familiar things from the likes of Tim Holtz, Claudine Hellmuth etc.
Make Your Mark is the only stamp store I can think of in York.
I don't know the Southampton area at all, I'm afraid.
Have fun on your trip!