The answer is, unfortunately, no. Due to airline regulations, currently there is no way you can fly into the the UK with a pet animal in the cabin of the aircraft unless it is a service or emotional support animal and your airline has a program in place to support them. This is a requirement of the airline. It is not a requirement of APHA or DEFRA (Department for Environment, Food & Affairs).
One alternative to avoid this requirement would be to fly with your pet in the cabin to Pairs, take a train to Calais and then take a ferry or take Le Shuttle across the Chunnel. There are several ferry services that serve the northern coast of France including P&O, Brittany Ferries, and Celtic Ferries (to Ireland). Remember, thought, that many of these ferries do not accept pets in winter months and most of them require that your pet travel in a car. Le Shuttle also shares this requirement. There are services that will pick you and your pet up in Calais like Folkestown Taxi and Pet Movers.
You can also consider flying to Amsterdam, taking a train to Rotterdam and taking Stena Lines or P&O to Hull. At this writing, Stena will accept foot traffic and, although the ride is longer, we hear it is very pleasant.
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