The classic Whitehave to Tynemouth or Sunderland is a nice ride but not really mountain biking. I’ve done it a couple of time on a cross bike. You could probably make it more technical if you wanted by picking your route through the lakes etc.
I believe the classic route for mountain bikes is Whitehave to Whitby. I’ve always been put off because of the logistical faff. Neither the start or finish are particularly easy to get to. They are not hard as such but it would add a day at each end for most people if you rely on public transport.
In true STW fashion I’ll ignore the question and give my own suggestion. HAve you considered the Sandstone way? It is logistically much easier. Park in Newcastle, Train to Berwick, Ride to Hexham then train or ride back to Newcastle. The route is good and there is some proper mountain biking in there. I mistakenly thought it was like the C2C and attempted it in a day on the CX bike (inc. riding hexham to Newcastle). It turned into a bit of an epic adventure arriving back in the dark even though we ducked out of a couple of offroad sections to speed up progress. Really nice route too.