Realistically if don't build it with all the bells and whistles, then what could it be built for. If you work on an assumption not going to try and get into Manchester area, but ending at new junction with existing line about 10-15 miles short, what price do you get, say £5-8bn
What advantages do you have, a route already surveyed, much of the land bought, often an excessively wide strip of temporary land. That is the key, you get to sell the unused & temporary bit back, and many who are disrupted don't want the long timescale so would be keen to agree to get things moving rather than years of inaction.
The other key is the bit about not delivering everything, just think if you make it more of a high speed only line, how much can save on earthworks by accepting steeper gradients.
My gut feeling is the section north of Crewe will go, instead Crewe will be reached from South as now, but will be bypassed and Manchester trains won't go via Crewe (plenty of existing trains go there from Manchester direction anyway). My reasoning is keeping it clear of built up areas makes it easier and cheaper to build, and if trains can get to south Cheshire at nearer 200mph, they will have gained enough time saving that a handful of extra miles at 80-110mph isn't a significant disadvantage.
My thinking is private money will sell off the spare lineside land to partly offset build cost, then simply charge tolls for its use as anybody else would.
Wouldn't even rule out some foreign money seeing it as extension of European high speed network. I am sure some suggestions of a few hundred meters of HS1-HS2 link and ability to process international passengers at Stratford or Old Oak could get it over the line (even if doing this in practice is actually harder). Can't rule out joining Schengen area in 2030s, especially if our current border scheme gets discredited for allowing migrants. Even ignoring borders there is plenty of money that see low cost trains as successor to cheap European flights, and worth investing in.