I have NO experience with bringing a cat from LHR, only JFK.
I bought my cat over in 2005. I don't think there have been any policy changes since then. But I also flew Swiss, and she flew in the cabin with me, in her carrier under the seat. There was NO way I could put her in the hold. She never would have made it alive, I'm sure.
In all, it went pretty well, but there were a couple of head-scratchers along the way. First of all, I understood that I had to pay a fee to carry her in the cabin, and at the check-in, they didn't charge me anything, despite my having told them I would be traveling with a cat when I made the flight reservation and bought my ticket, and then telling them again at the check-in.
Since she's a bit on the high-strung side, I gave her a tranquilizer and by the time we got to security, she was kind of groggy. I had to remove her from the carrier and carry her in my arms through the metal detector while the empty carrier was sent through the x-ray machine. God only knows what it would be like now (in the US) with those body scanning machines.
Once we arrived in Zurich, I had to, literally, BEG the customs guys to take her paperwork (health certificate, etc.). Also, it was my understanding that I was supposed to pay a fee (150 CHF, I think) to bring her into the country, and since the customs guys could barely bother to take my paperwork, they just waived me through.

Not that I'm complaining, but I was really surprised at how lax they were.
Anyway, I had been pretty stressed before the trip, but it all worked out fine.
Not sure if any of this helps, but good luck to you and your trip over with your fur babies.