I used a pet handling company to ship my cat UK to Singapore - you cannot book direct with BA/IAG. Obviously I can't ask the cat how she found the inflight service but the Heathrow guys were great and she arrived happy enough.

Shop around and study the quotes carefully, some include things like the pre-trip medical requirements or the import into the UK which you can handle yourself with some effort and save a lot

However this time I will be taking the route via Paris (Air France allow cats in the cabin in economy) and then having Folkstone Taxi collect us from CDG to drive us to Folkstone to be collected by a family member. For two reasons 1) cost, as I now have two cats and 2) Changi doesn't seem to have dedicated facilities and it is very hot here, prefer to keep them with me especially as the new one is a very nervous little thing!