The Government has announced that we are entering a transition period for Brexit. During the transition period, pet owners will be able to continue to travel with their pets in the same way as they do now (using a Pet Passport) until the end of 2020, as there are no immediate changes to the process.
Travel rules beyond this are yet to be confirmed. This replaces the previous guidance we were issued relating to a no-deal Brexit. The latest information is at https://www.gov.uk/guidance/pet-travel-to-europe-after-brexit