I do still have to check the instructions for Cramlington-Washington (be still my beating heart), but nae rush, I don't expect that to change, barring the appearance of a magical cyclepath from the south end of the tunnel to Fatfield Bridge.
Have you considered the Monckton Cycleway?
Easy access from the south end of the tunnel.
Some of the A frame restrictions have been removed, probably to improve wheelchair access, so reasonably swift progress can be made.
Restrictions aside, it's all tarmac or well made gravel track.
You would pop out just by Heworth golf club, then make a left (south) turn onto the A195 to cross the graded junction with the A194(M) at Follingsby.
Continuing on the A195 takes you, via a series of roundabouts, to Fatfield.
I suppose some may see it as less than ideal cycling, but it is mostly wide single carriageway, and is fairly flat, so you could bowl along at a good pace.
Nowt wrong with your proposed route, although it is narrower and may involve mixing it with car transporters and other HGVs going to and from Nissan.