Ultimately if money really was no object you'd build new faster routes. Pretty much everything north of Northallerton isn't really compatible with a high speed railway between the south and the north, so there's little point playing around with the edges with four-tracking and such like - just go the whole hog and build a new railway
After HS2 is built, I'd bypass York from Garforth to Northallerton following the A1(M) - controversial, yes, but York would still have a frequent service south, and I'm not sure it justifies being on a high speed route. I like
Gareth Dennis' idea of a new alignment from Northallerton to Newcastle via Teesside, and perhaps you could tunnel under Newcastle for a new station (or at least build a new bridge over the Tyne and expand to the south). After that you could then have a new route inland, diverging between Killingworth and Morpeth and following the A1, A697 and A68
Something like this, basically:
http://www.gmap-pedometer.com/?r=7340884 - this is a rough sketch I've just done so it's not meant to be indicative of whose houses or golf courses are likely to get ruined by passing Talgos. Also included this would be the reinstated Harrogate-Northallerton, Harrogate-Wetherby(-York), Castle Eden and Leamside lines which would be linked in to this, as well as an extension of the Borders Railway to Coldstream with a link to this route
Stations could include:
Wetherby Parkway (for Wetherby and Harrogate)
Northallerton
Teesside (Thornaby Yard)
Houghton Parkway (for Durham, Washington and south Sunderland)
Newcastle
Borders Interchange (Coldstream)
East Edinburgh Parkway
Edinburgh Waverley