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Not to put a dampener on things for you but given the experiences I have had getting bikes on trains (in Scotland, I'm in Glasgow) I wouldn't be satisfied with "No worries" over the phone.
Some trains such as the Glasgow - Edinburgh services have, as @h2o says, room for 4 bikes side to side however...(I believe) Scotrail rules are/were* 2 bikes per bike carrying carriage and I've been kicked off/not allowed on trains because the conductor wants to go 'by the book' regardless of what will fit.
Other trains have carriages with hanging space for bikes, 8 I think but again, I've had issues trying to get a booked bike onto one of these trains because the area was being used to store suitcases, prams etc and the conductor couldn't be bothered doing their job/asking people to move their luggage. Also had hassle from jobsworth conductors on these trains for hanging my bike from the wrong wheel!
*Could have changed since First Scotrail became Abellio Scotrail.
Thanks for all the input guys, much appreciated.
Thankfully the crisis has been averted. I managed to finally get hold of the sleeper and scotrail on the phone and after a bit of umming and ahhing from scot rail and a straight up "No worries " from the sleeper I've managed to book space on the train back to London....Phew