I've been there and you're right, it's really complicated. Here's what I've learned. First, you can fly a cat into the UK, but you must have it sent as cargo:
Pets usually travel as cargo, but there are different rules if you’re arriving by air or sea with an assistance dog.
This means that it will fly on the same flight you are on, but you need a separate ticket for the animal, one that is significantly more expensive than your own, and you need to fly into an airport that has freight handling. You also need to fly with an airline that handles freight, so low cost ones are out. Here is the gov.uk page listing the airlines that can do this and the airports they fly to that can handle cargo of this sort.
The cat will travel as luggage, as freight, and not be in the cabin with you. What's worse, although you might think it kind to drug the cat so it sleeps through its ordeal, we were advised not to do this because the people checking the cat on arrival won't run any proper blood tests, they just look at the state the animal is in. And if it looks like it's sick, for instance if it is waking up after a few hours on sleeping pills, they may quarantine it. Which will be no fun for anyone involved.
Your other option is to go by car or train. However, You can't just take it with you on the Eurostar. No, you need to take a special train that carries vehicles. Those are the only ones that carry "freight" and cats are apparently in this category. More detail on that in this gov.uk page about sea and rail routes.
In the end, we chose to pay the ~500€ ticket and fly the cat from Athens to London as cargo. We delivered it to the airport, it was taken away, and then on arrival we had to go pick it up from a special place in Heathrow. On the bright side, the cat was actually fine. It was obviously unhappy but not scarred for life or anything, and it was its normal haughty self a couple of days later. So it is probably less traumatic than it sounds.
My advice to you is to ask your airline. Tell them you need to send the cat as freight and ask them how much such a ticket would cost. And make sure you carefully read through every step of this step by step guide to bringing your pet to the UK: https://www.gov.uk/bring-pet-to-uk