Best to hire a company to do it.
I had a foster lady looking after Penguin while he waited for his 6 months (for US to UK they don’t have to go into quarantine but there is still the 6 month time period from the bloods/tests) to be up, the one company I hired got him from the fosterers house and to BWI airport in the US, another picked him up from Heathrow and drove Penguin to our house near Leeds. Poor thing spent a good 15 hours in the crate from start to finish.
I spent roughly £35 on the crate that Penguin was shipped in and I could fit all medium sized dog in it, it won’t even fit in our VW Polo.
Still, he got here safe and sound and the driver from Heathrow said he was the best cat, not a peep out of him. The guy was also thrilled to see the reunion as he normally takes animals to the airport.