My pets flew without me, but the paperwork stayed with them at all times. I believe DEFRA needs all the original paperwork with your pet, so perhaps if you want a backup, you can carry copies on your person.
DEFRA and British Airways were both really good with both my boys, though. (They taped my paperwork to my pets' crates, so there was no way the pets and paperwork wouldn't arrive separately.)
They do this every day, they're used to handling things.
