Despite the headlines, the "easiest" way to solve the A628 problems isn't in the Peak District, it's the bottlenecks on the western side of the hills - the problems between the end of the M67 and Tintwistle.
A Hollingworth by-pass would be a much better plan - no need to build in the sensitive Peak District, no need for an expensive tunnel. I've never been in any traffic jam in the Woodhead area, but I'd struggle to remember getting from Tintwistle to the M67 without some snarl up around Mottram.
Obviously a tunnel is more of a headline grabbing move, but it'd just get you into the Hollingworth/ Mottram gridlock (where the A57 and A6018 junctions are) faster without tackling that bigger problem.
I would suggest a third alternative:
Joe Public doesn't like lorries on the roads and would like to get rid of them, so move it by rail.
But Joe Public doesn't realise that (for the vast majority of such traffic) rail is not a practical. economic option.
That's it in a nutshell.
The Great British Public want any scheme that'll take other people's vehicles off the road (with the assumption that prices in shops won't rise because of this).
The Great British Public want any scheme that'll free up more road space for them (public transport is a great idea... for
other people!).
all the tunnels between Woodhead and Dunford Bridge now carry electric cables
If you wanted to build a road tunnel under the Pennines then I wouldn't slavishly follow the old British Rail route.
Since most of the road traffic will be heading from the "Lancashire" side of the hills towards Sheffield/ East Midlands (rather than north up towards Leeds/ Hull, which would use the M62 to cross the Pennines), the best route would come out in the Langsett valley (rather than the Dunford Bridge/ Penistone valley) - the
Little Don valley rather than the Don Valley in local terms - so that you are joining the M1 at J35A rather than at J37.
Even better, build it to Stocksbridge (where the current "fast" sections of road end), which'd leave the road through Langsett for more cyclists!