You probably need a bag of some kind for Eurostar. I cycled to
Waterloo, and when I started taking the bike to bits, a Eurostar bloke
told me that if I simply turned up a little earlier I would be allowed
to wheel it on.
Worth contacting Eurostar at Waterloo directly. If the reply is
favourable, get it in writing.
To send the bike, you're supposed to book it in 24 hours before and
pick it up in Europe.
I packed mine in a sheet of polythene, but they wouldn't let me take it
on. Had to put it in the "freight" section, where it was fairly well
squashed (luckily no damage). Others have said they simply carried the
wrapped bike on and put it in a luggage rack (there is normally room).