Some friends moved from Scotland to USA ~16 years ago with a whole menagerie of pets (rabbits, reptiles of numerous sorts, etc). I got in touch with them and their reply was "From what I remember of the process, it was expensive and involved some paperwork. I seem to remember that the plane ticket for the animals cost more than ours! We had to pack them up in a very specific way per the regulations. The animals flew British Airways, I recall. I think it depends on whether the animals are on the CITES list or not as to how they have to be documented." I notice you don't have BA on your list but numerous regulations have probably come in since then, and BA almost always codeshares with other airlines (from what I can gather), so I'm not sure how helpful this is likely to be to you, @Nan, but thought I'd mention it anyway.
Thanks, yeah, I'd checked BA early on. No dice.
The process to bring mammals into the UK is brutal.
You have to bring them in through only Heathrow, I think. (Unless you fly into some other EU country like France and then drive them across through the chunnel to save the money.) There are vet certificates, and all sorts of immunizations and test results that have to happen before they get on the plane. There used to be a 3 month quarantine once they got here - not sure if that's still the case or not - on top of that, and it had to be at an approved facility. (I had looked at moving over here about 20 years ago and had a pet cat at the time.) And mammals back to the USA... not as bad, as the UK is considered a "rabies free" country. Most airlines require a health-check, but that's about it.
CITES shipments are a whole 'nuther level of complex, yes. (Amazing what I've learned about moving animals around the world while trying to find out how to get our fish home.) And then there are shipments of animals of all varieties via an EU country. Where EU regs kick in.
The big problem with sending them "cargo" is that while "pet" fish theoretically should just be able to sit in a shoebox under the seat in front of me in the cabin, with no paperwork or intervention required, "Animal Cargo" has all sorts of requirements, including Vet inspections, special paperwork, etc., etc., etc. It's the pet-shippers/cargo company that jack up the prices, not the airline. The cost to have a pet in the pet baggage hold is a couple of hundred pounds, pretty much, across the board. But once it becomes "cargo".... £1,300.
Now, if I lived near an Air Canada facility I could take the fish directly to the Air Canada freight center and drop them off. And if there was an Air Canada facility in the USA near when we'd be I could go to that place and get their paperwork, take it to the US customs office, get it stamped, and then take it back and pick up the fish.
Unfortunately, there's not an Air Canada freight facility in the UK, apparently. And the closest one on the other end is Toronto. Which I'd be willing to fly into, pick the kiddies up, and hop an Amtrak down to the States and carry on from there.
So, still trying.