You will still be able to travel to the EU with your pet after Brexit and Eurotunnel Le Shuttle will still be the best route for taking your pet abroad or into the UK.
Under the current European Union Pet Travel Scheme, dog, cat and ferret owners can travel with their pet between the UK and other EU countries if they carry a valid pet passport.
If the UK strikes a deal with the EU, the situation for travelling with pets after Brexit will not change.
In the event of a No deal, however, the British government has issued advice on the measures that pet owners will need to take before starting their journey.
Should the UK become a third country there are three possible outcomes for pet travel. All the information you need can be found here, on the Gov.uk website.
Essentially, there are three options:
Listed – Part 1
Should the UK become Part 1 listed, the rules on pet travel will barely alter from the current system – the existing timescales and health preparations in place under the EUPTS will remain virtually unchanged.
Listed – Part 2
If the UK is given Part 2 listed status, there would be some extra requirements for travelling pets and owners post-Brexit. These would require an additional visit to the vet and some additional papers to be carried but would not prevent you from enjoying your trip.
Although this is the same procedure as the current EUPTS, under Part 2 Listed status this document would be valid for only ten days from the date of issue for entry into the European Union, and for four months’ onward travel within the EU. These health certificates would need to be issued prior to each individual trip from the UK to the EU.
Under Part 2 Listed status, owners travelling with their pet will also be required to report to a Travellers’ Point of Entry (TPE) upon arrival in an EU country.
If the UK is given unlisted, third country status, by the EU from 29th March 2019, owners who wish to travel with their pets from the UK to EU nations will need to discuss their specific preparations and requirements with an Official Veterinarian at least four months prior to their desired travel date.
Thank you to https://www.eurotunnel.com/uk/brexit/how-will-brexit-affect-my-pet-passport-/ for the information in this feature