Thanks for the suggestions. We're actually spending the night with friends in Newcastle on the way up. Modelsport looks interesting; I'll bring up Otley as a lunch stop idea. And I'll look for Wonderland Models in Edinburgh.
It sounds like a lot of your little hobby shops have either evaporated or been gobbled up by giant corporate monsters, like ours have... that's too bad.
What I'm really after is one of those little Mardave 1/12 scale kits, but I don't see them listed for sale in any shops. Does anyone stock them, or should I just mail-order one and tell people I got it when I was in England? 
I used to live in Newcastle, moved away just under three years ago.
By the time I left there were no "really" good RC car shops. The few that were good had either significantly reduced their stock or closed down - it's a tough market. Modelzone in Newcastle is a chain store as mentioned, only has a handful of kits, no spares or upgrades. Normally good stocks of PS spray paint though!
In "The Toon" itself there is a shop called the County House which used to do RC but cut back its stock dramatically to focus on the airguns. There might still be a bit of stock left. There is a shop in North Shields (10 miles towards the coast) called Phills Models who has been established a long time, kept some car stuff and has quite a lot of old stuff hanging around too. North Shields itself is not very nice but the Fish Quay is famous for Fish and Chips (although I don't think they are as good as most of the in-town chip shops). Tynemouth is nearby and is very nice with great walks along the coast. There is a shop in Whitburn called Whitburn Hobbies (which is actually on the other side of the Tyne between South Shields and Sunderland) that kept some car stuff as well. Whitburn is pleasant but very quiet. The other shops in the area kept very little car stuff but you never know what you might find!
The best car shop for browsing nearby-ish is in Middlesbrough - Cleveland Models. The owner carries a lot of Tamiya cars and parts, probably one of the best stocked for Tamiya in the UK! The shop is barely bigger than a broom cupboard. Middlesbrough itself is as pleasant as the smog which hangs permanently over it!
Although I've been to Edinburgh (a very nice day out) I've never been to the model shops there. Wonderland Models list a lot on their site but they don't keep all of it in stock - they are one of the breed of internet traders who list what their suppliers sell, rather than list what they have in stock.
Mardave stuff is not widely stocked, just a few mail order suppliers and the shops associated with clubs that run them.
Hope that helps - enjoy your trip!