The UK has requirements for bringing pets into the country which, while stringent, are fairly easy to meet. As long as you have all your pet's documentation in order, it will not be quarantined or refused entry.
The following is a brief summary:
If your pet already has an EU pet passport, then you need do nothing but keep rabies booster shots up to date. This information will be recorded in the EU pet passport, and your pet can travel with that as its documentation.
If your pet does not have an EU pet passport:
The general requirements vary depending on whether the pet comes from the EU (or a small list of non-EU countries) or from an unlisted country.
In common, though, all pets must be microchipped, have a rabies vaccination administered after the microchip, and have documentation of same available when they enter. When this is done, EU veterinarians can record this information in a new EU pet passport. Pets from outside the EU should be accompanied by whatever documentation the veterinarian is able to provide.
Dogs must also be treated against tapeworm.
The requirements vary a bit depending on the origin of the pet. Pets coming from the EU or listed non-EU countries have a 21-day waiting period after vaccination before they can enter the UK. Pets coming from unlisted non-EU countries must also have a blood test to confirm the rabies vaccination was successful, and a three month waiting period after the blood test before they can enter the UK.
Most pets will have their documentation checked on arrival in the UK, but those traveling Eurotunnel will be checked at boarding (except service animals, which are checked on arrival in the UK). When you arrive at the UK, if you arrived from outside the EU you and your pet must go through the red customs channel.