*Bakewell Speculation (Sorry)
In regards to Bakewell, I'd probably guess that it would run as a local service, rather than an express one. Still, it might be faster than the existing journey and passengers to Derby would probably enjoy not having to connect in Sheffield, the only place on planet Earth closest to absolute zero. Probably to get the best speed out of it, reduce the stops on one of the existing Buxton stoppers and then have that 1tph go down onto the Bakewell Line towards Derby. Once on Bakewell, you'd probably only be looking at stopping at Bakewell, Rowsley, Darley Dale and then Matlock.
So taking out Davenport, Woodsmoor, Middlewood, Disley and Furnace Vale, I guess you could conservatively cut down Manchester-Buxton travel times to 45-50 mins. Remember, these stations would still all be served by a stopper every hour. But keeping them in, it would be 56 mins, as usual.
30 mins Buxton - Matlock (Based on it being 20 miles - approx the same as Matlock-Derby, but with less stops, so it may actually take less than this.)
33 mins Matlock - Derby
=~2hours
Best time changing at Sheffield to Derby I've seen quoted is 1hr56. So, this would meet my conservative estimates with all stops on a local service. But, you could probably knock 10-20 mins off the journey times between the two if you were willing to do some more express running on the alternate train.
You know, the more I look at it, the more reinstating Bakewell seems like a great idea. It's not too bendy of an alignment - certainly no HS2, but definitely good enough to run service at a decent 60mph or so. There's 5 miles out of the 20 of functional track in the form of Peak Rail and a Freight Alignment coming out of Buxton. Nothing much built on the alignment either. But even ignoring the Derby - Manchester connectivity, for the towns in the middle - especially Bakewell - this would be absolutely game changing for them. Derbyshire has a surprising lack of connectivity between its North and South, so it would make sense to re-instate. I think that would be better than diverting one of the Sheffield trains straight southward at Dore.
*End of Bakewell Speculation
Hope Valley capacity does need to happen, I think it's probably more realistic than re-opening Woodhead. In regards to long term connectivity between the two, just boring a big tunnel under the Pennines isn't completely infeasible. It's been suggested for a motorway, although obviously it could be cheaper, higher capacity, safer and more eco-friendly if it was just a railway. Woodhead is quite a slow alignment, as much as those tunnels are quite juicy - but they are currently being used for National Grid, which would pose some obstruction, although I'm sure as others have said - not insurmountable.
Unfortunately, a proper tunnel is unlikely and Hope Valley capacity has been rumoured for what, like 5 years now? Still nothing has been done. It is absolutely ridiculous. Was literally an extra platform and a passing loop.
Sheffield also needs a similar job as to what was done in Derby with the remodelling there - the work at Derby really was fantastic and the improvement in speed and reliability is noticeable!