Yes, I learned to ride at Penshaw many years ago. From what I've heard it hasn't changed all that much. Average riding school, has all the usual accreditation. They did have a small cross country course there but its not always usable because there tends to be horses turned out in those fields. Nice view of Penshaw monument, an Acropolis-like thing on a hill (!)
When I was there it was a fair mix of horses:- a couple of cobs, big and small ponies, a couple of TBs and TB crosses.
Hacking out is possible too.
But yes, I did find there was a 'pot luck' element to lessons, particularly instructors changing. My current school (not in the NE) is of a similar size and aimed at a similar market, with similar horses, but I find the organisation miles better.
I believe Northumbria Uni students used to ride at a school north of Newcastle, I'm not sure where. You might want to ask their student union.