On saturday I got a turn-up-and-pay ticket for about 23 quid, took the bike with no reservation to Doncaster and then managed to get a bike space booked at Donny from there to York on the East Coast train. I've never tried without a reservation, but have been told elsewhere that East Coast guards won't let you on without a booking.
Had I not managed to get on the East Coast train my contingency plan was to wait for a Northern train service to Leeds, then change there for a local service to York - this would have been a lot slower though. I also looked at catching the local service from Cleethorpes back up to near the bridge (I reckon that after 67 miles another 30-odd back towards the north would be, erm, 'challenging' for me) then cycling over it and going to Hessle/Brough but it worked out as expensive as the Doncaster option.
(I'm probably going to cheat and get my little sis to come and rescue me in the car!)