Possibly because I used to run a Morgan, someone asked me the other day, if I knew of any companies offering self drive classic/sports car rental based in the Northeast.
Have you tried contacting the classic car restorers in Wylam? I'm sure they would know most things classic car related in Northumberland and may be able to point you in the right direction.
good thought, I suspected that beamish morgan might do morgan rental, but dont see anything on their website, I think my pal was probably thinking of something else anyway, something like an e-type or big healey, or possibly a noisy TVR