This may or may not help, but I was trying to figure out how to move some tropical fish and a hamster back to the USA last year. I phoned/emailed a very large number of airlines and the bottom line was that only Air Canada would allow me to take the pets to a cargo terminal and have them shipped to the USA without a middle-man pet shipping company involved. Unfortunately, there was no Air Canada facility in Scotland and no way to get to one from there. There was a Spanish airline that would carry them (in the cabin, I think), but we'd have to have traveled overland to Spain to connect up with the transatlantic flight, since they couldn't fly from a UK airport due to UK regulations. It was primarily the UK regulations on flying animals in/out that were throwing a wrench in things. I will try to find the list of carriers I contacted and the outcome - I think I posted it somewhere on a fish hobby board - and will post it here if I do.
I also think I remember finding that if I was trying to do the reverse, the UK had extremely strict requirements for mammals from the USA to go into a special facility, and that flying into the EU and then taking ground transportation was much easier. But that was back when the UK was part of the EU. God only knows now.
EDIT: Here was the final list, if it's of any use -
American - codeshare with BA. Requires a pet-shipper.
Air France - dogs and cats only. Pet Shipper.
Aer Lingus - As cargo, only to major international destinations where they fly their largest aircraft (no puddle-jumpers). Not sure about having to use a pet-shipper or not for dogs/cats.
Aeroflot - no
AeroMexico - only pets on flights of six hours or less
Agean - fish ok as pets in hold, don't do cargo to the USA.
Air China - cats and dogs only, pet-shipper.
AIr New Zealand - pet shipper.
Alitalia - dogs and cats only. Don't know about pet-shipper or not.
ANA - no fish
Brussels Air - cargo. pet shipper
Delta - cargo. Requires a pet shipper.
El Al - They allow birds in the cabin. Not sure about cats/dogs.
Finnair - They allow rodents in the cabin.... Have to go cargo and require a shipper for international.
Iceland Air - no
Japan Airlines - pet shipper
KLM - only cats and dogs, codeshare Delta. Pet Shipper
Korean air - no
LOT - website unclear
Lufthansa - pet shipper
Norwegian Air - no pets.
Quantas - pet shipper
Swiss AIr - pet shipper
Transat - dogs and cats only
TUI - dogs and cats only.
SAS - no
Singapore Airline (flys out of Manchester or London) - they take fish as "cargo pets" - that translates to "cargo" requiring a pet shipper.
United - cargo. Requires a pet shipper.
West Jet - no
The cost to ship a betta fish and some $1 corys, plus the hamster, across the Atlantic was closing on £2,000.