Don't forget all, you can book any journey with any operator, as someone on here has already mentioned I think (using Southern to book London to York?)
East Coast is very good and allows you to book your bike at the same time as your rail ticket, it can be a bit slow and suffers from a few crashes from time to time but bear with it and you should be ok.
Some points to note:-
1) Book a return journey as 2 singles will work out a lot cheaper
2) Opt for fastrack ticket collection were you collect your tickets from the machine at most UK stations, I once waited for some posted tickets which never turned up and ended up not making the journey and been unable to claim as the refund cost would have gazumped the ticket cost.
Hull Trains are also very good but you will have to speak to someone in their sales office to book your bike so you might as well book rail travel and bike at the same time. Their office is in Plymouth, Devon by the way and not Bangalore, which is nice...(as you can understand them)
www.eastcoast.co.uk
www.hulltrains.co.uk
Some people have asked about the return journey and looking online tonight there are plenty of trains leaving Cleethorpes Sat 19th June to Kings Cross, usually with one change at Doncaster - 8 trains between 05:18 and 11:13 single fares between £16.45 and £22.45, alternatively why not prolong the enjoyment / agony of your stay oop north and ride 35 miles back across the Humber Bridge and enjoy a personal guided tour of the mighty city of Hull by yours truly before catching a train home in the afternoon. Approx 3 trains an hour Hull to London (most change Doncaster as well) single fares from £10.00 to £18.70...