Regarding the suggestions of tunneling the whole of HS2, the tunnels create a lot of emissions during construction so from an emissions perspective it's better to avoid tunnels.
Construction emissions are an irrelevance in tunnel construction.
Consumption of concrete/metal/energy for the TBM pales into insignificance compared to the emissions from burning carbonaceous fuels for energy.
If tunnelling HS2 gets more public transport use, tunnelling HS2 is certainly the green option. Indeed it only has to shift a tiny amount of transport of air traffic or other carbon powered traffic onto rail to be a net benefit, especially given the
very long asset lifetime of a tunnel.
Fretting over construction emissions of such schemes is rather like fretting that a project costs a few pounds too much, whilst spending hundreds of pounds a year trying to work around the expenditure.
If we are serious about decarbonisation, almost any fixed infrastructure is worthwhile if it eats into aviation market share.
Sustaining domestic aviation in the UK would require at least 930MWe of electrolysis running day and night, a synthetic fuels plant, and a large carbon capture infrastructure capable of atmospheric capture of ~1.5 million tonnes of CO2 per year.
And energy infrastructure to keep that going.... forever.
[This obviously excludes the flights to the Republic of Ireland]
Or you can build a bridge/tunnel and be done with it.
The crossing is technically challenging but certainly within our engineering capability if someone is willing to pay for it.