I am wondering which one is more commonly used in the United Kingdom: car hire or rent a car?
In British usage (at least up to the last few decades)
("hire" is not used for people: rather "employ" or "engage").
I qualified this because American usage is now recognised here, and I would guess is used by some people.
But "car hire" and "hire car" are both common expressions still.
I don't know about the UK, but I would suggest that "hire car" is a more equivalent term for "rent-a-car" than "car hire". I tend to think of "car hire" as a service, and a "rent-a-car" as a product.
In Australian English, at least, the more common term is probably either "hire car" or "rental car".
As an aside, I wouldn't be surprised if 'rent-a-car' turns out to be trademarked by a company, although a quick google hasn't immediately shown up any likely suspects.