To make any significant time savings between London and Edinburgh, you basically need to new alignments. The first from Northallerton to Birtley (avoiding Darlington and Durham), and the second from Cramlington to West Chevington (avoiding Morpeth, Pegswood and Widdrington).
Even at "only" 125mph, these alignments will be enough to get the London - Edinburg journey time under 4 hours as they will avoid all the twisty sections where speeds drop down to 75mph (50mph at Morpeth).
WOAH WOAH WOAH - back it up Pedro. Avoid Darlo? What is this of madness do you speak of? How will I get to the match?
What kind of route do you propose? I am thinking about the area between Northallerton and Newcastle, that I know well, and struggle to pick a decent route.
I suppose you would go straight on at Northallerton, branch of the existing alignment somewhere north of Danby Wiske and aim for the gap between Darlington and Teeside Airport. You may end up having to cross the Tees twice as the river is twisty at this point. You have a very small gap to aim for. You cant go any further left as you then run into the airport and then Teeside.
I guess you would then aim for a route on the axis Teesside Airport - Stillington - Sedgefield - Trimdon - South Hetton. Then you run into a problem as the open country starts to run out. The issue is how you then swing back to the existing main line through the Wearside / Tyneside sprawl. I suppose you could run via the Leamside route and into Newcastle that way but that route has a massive dog leg at Victoria Bridge that will slow down the trains.
Alternatively you could branch of the existing alignment at Dalton on Tees and pass under Darlington. You will need a route over both the A66M, A1M & River Tees before turning north. Your axis here would be, I suggest High Coniscliffe, Middridge, Spennymoor, Tursdale- Leamside.
Both routes are hilly and would require extensive tunnelling operations; both routes are through former mining areas and would need lots of careful preparation to ensure there were no subsidence issues or undermining problems. The above suggestions take no cognisance of the topography and geological makeup of the area. You should also consider the potential for the planning unpleasantness a proposed new route would bring. The area you could force a new line through would be very limited. To far right is Teesside and the coast, to far left are the Yorkshire Dales
It seems like an awful lot of work for a very little payback. It might be possible in a bigger HS2B scheme but on its own the costs would be astronomical!