I believe the old line ran south of Scotswood Rd (through what is now Newcastle Business Park) and crossed over it approximately where the junction with Whitehouse Rd is today. That section north of Scotswood Rd is used today as a footpath/cyclepath (NCN72) - although the formation has been broken in two by the A1 western bypass.
There may be room for tram tracks between the Business Park and Scotswood Rd though.
Looking at the map though its crossing from Gateshead so are they going to build another bridge or go round and use the old rail bridge next to the road bridge then go straight up - Cant see anything on the map thats going to come from Elswick along to Scotswood then up. It goes Elswick > General.
Without reading this more Im guessing they would have such service on a loop in each direction?
Street running? Nah not in the West end, would never last.