I used to quite like the idea of building a new station in the vicinity of Sheffield Victoria, my personal preference later superseeded to significant sections of new line and tunnel to the West of Hope!
A big part of the difficulty getting from Sheffield to Manchester is the complex geology of the South Pennines. Mam Tor, the shivering mountain, is the best known but as you travel along Edale you can see several old land slips. We know the tracks were sliding between Strines and New Mills very recently. Cowburn Tunnel is very wet as is Totley Tunnel. Minor land slips along the route occur from time to time and this makes it difficult to raise line speeds at reasonable cost.
In Norway they just blast their way through mountain ranges with minimal lining. Tunnels through the Pennines go through more difficult rocks and need different methods usually requiring lining throughout. In an ideal world, money no object, I'd start boring below Sheffield Midland, almost dead straight to Manchester Piccadilly, totally in tunnel, virtually level. No junctions, no bridges, level crossings, trespassers or weather to contend with. 20 minutes end to end for 32.5 miles.The Sheff-Man or Man-Sheff line - the Peak Tube. As a concession I might allow my friends in Glossop a deep tunnel station as the straight line goes almost exactly beneath Glossop station. Start with 3 trains an hour, one stopping at Glossop.
It'll never happen but imagine getting between the centres of those two cities quicker than to their suburbs! A good example of railway levelling up in practice.
Dream on - but see how well the Elizabeth line is doing and look back at the difficult times it had to reach completion.