Are there any programs that allow library books to be checked out, as ebooks, specifically in UK?
In the UK you can (I suspect it depends on the county/borough) borrow ebooks from your public library.
They use Overdrive as I suspect many others do, so I think only do ePub readers(and only those that understand Adobe DRM), so no Kindles. Overdrive does do Kindles but only in the US.
See the entry page and you can browse without registering for the London Borough of Croydon
The Amazon Kindle library exists for amazon.co.uk.
You need to pay to sign up for Amazon prime