With the GCR rebuilding (another of my favourites) you are starting with a blank slate. The GCR was laid out very well, and I'm certain it could sustain 155 mph on the old alignment. There was no shortage of tiny stations (Finmere, Quorn and Woodhouse, etc) but there is probably no need to reopen them. Going for a limited-level reopening, 125 on double track for much of the length with Chiltern extending its service to Brackley or Rugby, I reckon you could reach Sheffield in about 105 minutes, Nottingham in about 70 minutes and Leicester in about 55 minutes. By diverting some Cross-Countries as well, you could probably cut 30 minutes from the Reading-Sheffield time by avoiding Birmingham.
Routing HS2 that way (always my preferred option) would bring Nottingham down to 45 minutes and Sheffield down to 60 minutes, mostly by avoiding stops and running a 6 tph each way frequency. This includes one to Edinburgh via Rotherham and Leeds, one to Glasgow via Woodhead and Manchester, one to Bradford via Penistone and one through train from Paris to Glasgow, leaving two 200 mph semi-fasts. The Rugby-Leicester-Nottingham section would somehow have to squeeze in two 125 mph paths for Cross-Country. Nottingham-Sheffield would have four tracks to work with, allowing it to accommodate a Norwich-Manchester-Glasgow service. So would anything south of Rugby, to accommodate suburban services. The Woodhead would have to accommodate a diverted TPX South route as well, but would be a 125 mph route rather than full 200 mph LGV.
If this happened, I would expect Midland services to start adding extra stops (no way they could compete on speed). However, the MML would be moving more passengers than ever, to feed the GCR and international services. Tweaking the MML paths to allow for this might make freight pathing easier, so allowing more to run that way. I would also expect a major boost to the economy of the region from companies relocating for the good transport links to the Continent as well as London and the North.
Its doable, but would cost a lot of money. However, probably not as much as an all-new HS2.