There's
Arbor Timber at Langley Moor near Durham.
There's nothing machined until you ask for it so it's a matter of browsing, picking and then waiting for your piece(s) to be machined. That may take a few days. If you're not that picky you could order online and would probably be OK - I've had no problem with the quality of their stuff.
The staff can be a bit offhand though and it often takes ages to get through on the phone. I once phoned for some 32mm MDF and was told no problem, just come through and pick it up. When I got there they looked at me as if I was crazy. "32mm MDF? We don't keep that on stock. We'll have to order from the warehouse". Also, when you go to the shop they often have a competition to see who can find anything to do other than serve a customer.
What can you do though? There doesn't seem to be any alternatives around here.