MTB vs Hybrid/Tourer/CX vs Roadie
I'd say the MTB will be most comfortable for the trail, but may not be as suitable if you want to do a luggage laden tour rather than a day ride on part of it. A Hybrid/Tourer/CX is more than capable as long as you upgrade the tyres to say 32s or higher. There are a few short sections around the Warrington/South Manchester area that can be muddy and when I did it on my 32c my biggest issue was mud getting jammed up inside the 'guards. Increasing the clearence would solve that.
A roadie just would be uncomfortable, sections of the trail are tarmacked but its old, cracked and inconsistant.
The only section I had to walk it was up the pennines themselves - the trail above the woodhead tunnels is steep and rocky, a fitter person could perhaps stubborn themselves up but I felt it was better to push it up that mile or so.
The sand issue from the gravelly surfaces they like to use meant my entire drive was coated in crap at the end of day 1, next time I'd be taking a rag and lube to do a clean on it.