We live in South Oxfordshire. So our train routes east include starting at Oxford, Didcot or Thame and Haddenham. And there's a coach option.Yes. I want to travel by train from Bath to Norwich to ride the Rebellion Way, but I know from bitter experience the chances of doing this without problems are very low. Reserving a bike space means nothing, because of the issues already mentioned.
Sticking to GWR routes... we've often taken two full-size solos, and occasionally a tandem and a solo to London. Often with full touring and camping gear.
So from Bath I'd go: train to Paddington, ride or Crossrail to Liverpool Street, train from Liverpool Street.
The cycling route across London is now OK, and here are the restrictions for bikes on Crossrail:
https://tfl.gov.uk/modes/cycling/cycles ... -transport
The booking and accommodation aren't as good as I'd like but I wouldn't let that put me off.
Jonathan
PS: If you're not used to travelling by train in the UK remember to look for suitable railcards.