I take the London train all the time - lovely big shed-like carriage for bikes, no need to book.
You don't have to book a bike on but if there are a few of you going it might be an idea to book them on. It's free, you just have to call up and tell them what train you're getting.
If there isn't space in the bike section and you have it booked on then they will try to squeeze it on, if you haven't booked it on you'll be left on the platform waiting for the next train.