United Kingdom
Bikes are welcome on most trains all over the place, but as always it’s bloody complicated and there are exceptions.
Most local trains that stop in every town along the track will let you just roll on with a bike fully formed, and you’ll take your chances as it’s first come first serve. You’ll probably be lurking awkwardly in the doorway. Big-ass trains like the ones going from Bristol to London have a special bike storage carriage at the end of the train, and you might need to reserve a spot.
Seeing as the UK is a myriad of local private train companies, there is no one website you can look at for consistent rules, but The Train Line took a good go at it.
See if you can reserve a bike on that line, and if there is any uncertainty you can usually book it and talk to them on chat.
They’ll either say “this is fine”, “we’ve reserved you a spot”, or “nope you can’t take a bike on this, you fancy a refund”?
International Trains
Read about Eurostar.