An earlier post (click here) referenced new types of bus layout which can accommodate your (normal size) bike - here are a couple of snaps of how the system actually works. The bus in question is the X46 service from York to Hull. The service you should take is the X46, not the X47 btw, which only has provision for folding bikes ....

.... and from Chinley, you can travel exclusively on Northern trains via Sheffield and Leeds to York on trains with reasonably good provision for cycles.

Here's what the bike storage area looks like (here)  and  (here)   and (here )