Most local libraries in the U.K. now offer digital audiobook services.
The audiobooks available from local libraries are commercial audiobooks and a limited number of copies of each book are available for loan. You will first need to check the availability of your chosen book and then you will be told if it is available for immediate loan or if you will need to wait for it to become available. Your loan will be for a specified period and it may be possible to renew the loan if you don't finish the book in time.
Local libraries use a range of apps to deliver their services and you may find some of them a little awkward to use. If your library supports more than one app it's definitely worth trying them all and offering feedback on any issues you have.
The Leicestershire Digital Library Service webpage gives full details of the service available in Leicestershire. Audiobooks are available from two sources, BorrowBox and R B Digital. I think you will find both services accessible to VoiceOver users, but you may experience a few issues. You will need to install both the BorrowBox and R B Digital apps to have access to all the available audiobooks. (Note that there are no spaces in the R B Digital app name. I have inserted spaces here so that screen readers are more likely to speak the name correctly.)
Leicester City Library audiobooks are offered via BorrowBox and Overdrive.