The Pet Travel Scheme is a UK Government scheme that allows pet cats and dogs to enter the U.K. without spending time in quarantine.
In order to comply with these rules you will need to do the following (In Order).
Have your animal fitted with a microchip.
Have your pet vaccinated against rabies.
For qualifying countries, after a period of 21 days following your pet's vaccination, your pet may then travel to the UK.
If you have any evidence that your pet is fit and healthy (like a letter from a vet stating that your pet has been recently examined and what the findings were), then this would be helpful.
Between 1 and 5 days before the date of departure your dog (this does not apply to cats or ferrets) must be treated against tapeworm. You must have this treatment completed by a vet and ensure they issue you with a certificate to this effect/stamp your pet's passport.
Please be aware that there are different rules applicable determined by where your pet has travelled from. If your pet can easily be identified (because thet have been fitted with a pet passport microchip) then importation will be much easier. Always try and travel with your pet so that you are on hand to answer any questions as they arise.
There are other considerations you will need to take into account before bringing your pet back into the U.K. for instance the route that your pet travels must be approved by DEFRA. The airline that you use must be an approved carrier and the country that you are flying from needs to be a qualifying country.
The above information should only be used as a guide, as the regulations are constantly changing. A full list of regulations can be found by using the link below to the DEFRA website.