To leave and re-enter the UK with your cat or dog, you must have a pet passport. This can only be issued by an official veterinarian; this is an additional qualification gained after graduation and not all practicing vets are able to issue passports.
For the majority of European countries, the cat/dog must have a microchip and a rabies vaccination, and the passport can then be issued. You must wait 21 days before leaving the country, so it is important to arrage your passport in plenty of time. For some European countries, and those further afield, the regulations are different or there are additional requirements. It is the responsibility of the owner to ensure all requirements are met.
We recommend https://www.gov.uk/take-pet-abroad/overview as a good starting point.