Eurostar doesn’t accept any animals on its trains except guide dogs, which travel for free.
Guide dogs can only board at London St Pancras and travel to either Paris, Brussels or Lille.
Once on the continent many trains do accept pets, so one alternative would be to catch a ferry to the continent, then take a train onwards from the port.
Different trains have different policies regarding pets, so it’s best to check before you book which trains do and don’t accept them. Find out more about Routes and Trains