I would start from Stratford International then head north along what is basically the line to Tottenham Hale. This can be done on viaducts of simple cheap construction I think. If it is tight between there and the M25 then there is a possibility to follow the river.
HS to Birmingham
The line would turn west at Turnford fitting through a gap in the housing and out into the open fields there.
The route would be bent round the south of Hatfield before picking up the A414 alignment to the south of St Albans then following the M10.
North of there along the M1 round the north of Redbourne. North West through open country passing south west of Whipsnade
This then hits the WCML at Linslade. Here the line would cross over the WCML and continue pick up the A4146 before having the option to join the WCML south of Bletchley.
4 tracking from Wolverton to Birmingham New Street might make a high enough speed possible.
If not then the line would bend westwards again to go to the west of MK. There is a lot of open countryside here so the next notable passing point is Silverstone Circuit. The line would pass East of here and then to the North East of Silverstone itself.
There is a lot more countryside, but the line would pass North East of Woodford Halse offering potential to join the GC alignment if needed. I do not think it would be however.
Looking at google it is possible the GC alignment heads NW at this stage in which case might as well use it. The next landmark to pass is Draycote water, which the line would pass to the south west.
Now we are approaching Coventry so a further turn west is needed to bring the line across the south of Coventry and in to Birmingham International
HS to Everywhere Else
From the Turnford junction the line would continue to Stansted before curving up to pass Cambridge (I believe an important high technology centre) to Peterborough where it would fit along the ECML to Stamford, continuing North West where the ECML begins to turn northwards (upgrades from here provide potential for Newcastle)
There are issues here, one option is to swing West passing north of Melton then following the Old Dalby alignment towards Nottingham and then veering further west to pick up Toton.
The other option is to ignore Nottingham and carry on to Retford.
The ultimate aim is Sheffield Victoria and Penistone (ish) where a triangle takes trains to Leeds/Bradford and Manchester.
London End
International trains go to St Pancras and reverse
London trains terminate at Stratford International* for connections into crossrail etc
London Liverpool Street can be used for some services as well with capacity having been freed by crossrail.
Advantages
Connects major airport development opportunity at Stansted, important business centres at Canary Wharf and Cambridge as well as the City and the main northern towns.
Releases capacity on the MML for more trains between Nottingham/Derby and Leicester to London.
Manchester gets a slightly quicker train to London but much faster services to places like Leeds, Sheffield and Cambridge.
Offers real opportunities for development on the old coalfields (may even get some funding that

)
Simple to connect in the North East and Edinburgh so has more potential beneficiaries
Birmingham line used solely for Birmingham meaning a lot more capacity and paths for MK stoppers.
Interchange with the WCML south of MK gives routing options
Disadvantages
Slower to Manchester than other HS connections.
No really obvious connection for Liverpool (unless HS continues through Manchester to Liverpool)
Harder to get to Birmingham
Other thoughts
Two things I have always said should be in HS:
HS should have a 6 platform station in Stoke on the current station site and this does not deliver that. This would serve as a looping station for the Manchester - Stoke - Birmingham International* - MK - London trains as well as allowing dedicated shuttles to places like Stafford and Crewe to link in to other services.
HS should also have a 6 platform station between Nottingham and Derby near Toton which again this does not deliver.
As I use the HS between Jiaxing and Shanghai a lot it is very clear how futureproof they have made that with the station at Jiaxing South how I would hope we will build the main intermediate stations. All platforms are off the main running lines and there are three platforms in each direction.