A reply from the British Library arrived this morning. If it looks a bit crumpled, then that's because the dog got to it before I did.
It basically states that they require a little more information before they can grant permission but it doesn't look like photography with a hand held camera will be a problem in the public areas (the use of tripods and additional lighting would require the supervision of a member of the library's press office).
With regards to the recordings, they again need to confirm what equipment would be used plus they need information about whether it would be for the interviewing of library users or "simply recording the sounds of the library". The latter response suggests that my request isn't going to met with confusion.
The letter also contains the email address of the Press Assistant requiring the additional information so I'll send my reply electronically and hopefully get everything confirmed.