In the event of a no-deal, pets would continue to be able to travel from the UK to the EU, but the requirements for documents and health checks would change.

If you wish to take your pet to the EU on or after 31st January 2020 pet owners would need to discuss preparations for their pet’s travel with an Official Veterinarian at least four months in advance of the date they wish to travel. Pet owners should keep an eye out for any further instructions issued by the UK Government.

For more information please visit - https://www.gov.uk/guidance/uk-nationals-travelling-to-eu-essential-information.

and:

https://www.gov.uk/guidance/pet-travel-to-europe-after-brexit

The UK Government has launched a ‘Get Ready For Brexit’ campaign- further details can be found here.