No airlines will permit you to take your cat into the cabin. Neither can you check them in with your normal hold baggage at the airport terminal.
The only way that you can transport a cat is to book it in as 'unaccompanied cargo'. (The fact that you'll be travellling on the same flight is irrelevant. You have to follow the same procedures as if you weren't travelling). This means that you have to deliver your pet to the cargo terminal at the departure airport and then collect it from the cargo terminal at the arrival airport. (Cargo terminals are often several miles from the passenger terminals and not served by public transport).
Budget airlines don't offer an unaccompanied baggage service, so you'd have to travel with a 'full service' airline. (Well, the cat would have to travel with the full service carrier. There's nothing to stop you from taking a budget flight yourself). Full service carriers heat their holds when animals are carried.
Air freight charges are quoted separately for each item carried, based upon the weight and size of the container. (i.e. there's no specific 'fare' for a cat) but they charges are never cheap. The relevant BA link is here: http://www.britishairways.com/travel/pet/publi c/en_gb
Dazf's answer made me check a few more links. I found this important information on FlyBe's site:
"Please note: In line with other European airlines during the Winter period, Flybe. will suspend the carriage of all animals between the end of September until the beginning of May each year."
So, if you're thinking of travelling in the near future, you won't be able to travel with your cat.
The FlyBe site does prove that I was wrong in stating that no budget airlines accept cats. FlyBe do so, but only on certain domestic routes and only on flights operated by BAe146 aircraft. (Other aircraft in their fleet don't have heated holds). As with BA, the procedure involves booking pets as 'cargo'.
I remain confident that it would be cheaper, easier and less stressful for the cat, to travel by train.