To travel with your bike on Eurostar, you need to book a space for the bike separately to your train ticket. To make sure it travels on the same train as you, book it in advance otherwise,  you could wait up to 24 hours for your bike to arrive!

Book your seat on the Eurostar and then book the bike by calling the EuroDespatch team on 03448 225822 (+44 3448 225 822 from abroad). The lines are open Mon-Fri, 09.00 to 17.00 hours excluding Bank Holidays.
 
The cost to pre-book ranges between £30 and £55 (as of Summer 2019) depending on how much in advance you make your reservation and whether your bike is fully assembled or boxed/bagged, they only have a limited number for assembled bikes so you may be offered a boxed option, where they provide a box for the bike.
 
Allow enough time to drop your bike off at least 90 minutes prior to any pre-booked service, and/or to collect your bikes from the Eurostar luggage office before getting any connecting train.
If your bike is booked on a service departing prior to 08:00 you would be required to drop your bike off at least the day/night before.
 
https://help.eurostar.com/faq/uk-en/question/Can-I-take-my-bike-on-Eurostar
Once booked you will be advised if you have to use a box or not,  allow another 90 minutes as you will have to personally dismantle the bike and put it in the box. wheels off and skewers out, handlebars off, saddle off, pedals off and put it in a Eurostar provided bikebox. They provide a bike stand and tools, but its best to have your own tools as they maybe a queue.

For further details on this service see the following links

https://help.eurostar.com/faq/uk-en/question/Can-I-take-my-bike-on-Eurostar

https://www.eurostar.com/uk-en/travel-info/travel-planning/luggage/bikes