I moved my two dogs from Perth, Australia to the UK with out am animal cargo company. It is totally possible, but there is a lot of paperwork. It saved us a nice chunk about $1,500. Pets flying out of Australia and into the UK must be in cargo. Get the most direct route if possible. We flew domestic to Sydney (which was $125 per dog), then picked up our dogs and filed the export papers in Sydney (stayed for two days to give dogs a break) and then flew direct to London. That way there was no risk of the dogs being left on a tarmac during a transit. You will need to get a rabies vaccination a certain time in advance, I can't remember what the timeline is. Other than that, it is just a vet clearance with in 3 days of their flight (if your vet won't do it with out a animal transport company, just look up "flight clearance vet") and during that clearance they will give the cats a dewormer. Your vet is either not certified to conduct the vet clearance or just doesn't want to deal with the paperwork. We used a vet that was specially certified to do this. 24 hours prior to the flight, you will file the export papers with the appropriate department in the states. You need to have their vaccination history, rabies certificate, and vet clearance. I am sure there are different appendix needed for the states, if you go a government website on animal export, there should all the info. Basically it is totally doable on your own, as long as you are organized. I would start out by calling the airline that you want to use and ask for their price on animal transport. They charge a cargo rate based on the crate size. At least they did in Australia, could be totally different for the states. Also our dogs are under 30 lbs each, so they were allowed in the same crate, if your cats like each other, see if you can put them in the same crate, so they have a companion for the journey.
You do not need a cargo company to collect them in England, if you will be arriving at the same time.
Side note, it took about 7 hours after we landed in London for our animals to be cleared. We were exhausted from our 24 hour flight and ended up sitting in a McDonalds nearby for 7 hours to wait for them. Where you pick up the animals is next to, but not in the airport. So be prepared for a long wait! But the reunion was oh so sweet.