ok, my two cent's worth.
There seems to be consensus that any HS network out to serve central London, Heathrow, Birmingham, Manchester, Leeds, Newcastle & Scotland. Yet one line to all those places would be so ridiculously roundabout as to be completely ineffective. So here's an idea
HS2 (West):
Existing proposed line to Birmingham, with spurs into Bham city centre & Manchester city centre. Line continues north from Junction with Manchester spur (around Crewe somewhere?) to head north to Preston, Carlisle, and then a Y-junction for lines into Glasgow & Edinburgh. This would give a journey set of:
London/Birmingham/Manchester-Scotland: Preston (for Lancashire & the NW), Carlisle (for the Borders)
London-Bham: Bham international only
London-Manchester: Manchester Airport only
HS2 (east):
Instead of going via Birmingham, a second branch to head due north once the line clears the M25, heading straight north to East Midlands parkway. There build a People mover to East Midlands Airport, electrify the lines into Nottingham & Derby for rapid acceleration 'metro' EMUs, an modify Nottingham station to accept classic-compatible HS stock.
From EMP, carry on due north, skirting west of Long Eaton, before heading north, approaching Sheffield from the East to a reopened Sheffield Victoria. From there continue north, curve round Chapeltown to skirt Barnsley to the west, then head dead straight east of Wakefield till you reach the southern approach to Leeds. At leeds use the existing Pontefrac/Hallam line approach to a station on the current Crown Point industrial estate, then tunnel under the river, for about 4 miles to an emergence at Shadwell. From there head due north to Tees Valley parkway (rapid transit on the Tees Valley line for Darlington & Middlesbrough), then on to Newcastle, and finally, a route cutting through the Cheviots to Edinburgh.
Journey set created of:
London-Newcastle: East Midlands Parkway, Sheffield, Leeds, Tees Valley Parkway
Nottingham-Edinburgh: Sheffield, Leeds, Tees Valley Parkway, Newcastle
London-Edinburgh: Sheffield, Leeds, Newcastle
as the western line is probably faster, there may well not need to be a London-Edinburgh service, but it would serve as a relief line and would link all the major cities on this alignment.
Finally, Heathrow
Heathrow is miles out of the way for any lines going north from Euston. But it is perfectly placed for lines going West & South out of Paddington (in my opinion, the obvious terminus for a Western line serving Reading, Bristol & Cardiff)
It is also very well placed for an 'International Connection line', running round London to the south, calling at Heathrow on a N-S alignment (as opposed to the Paddington line's E-W alignment) before heading north & joining HS2, allowing Scottish services to the continent to run south, have a call at London, and get to the Channel Tunnel without having to tunnel through most of London.