There's nothing to suggest that it's only open to UK residents. But you can call to check: +44 (0)20 7412 7831. There's also a form to fill in on the same page. When you've filled it in, you send it to:
Sound and Moving Image Copying Services
LGF, BLCC
The British Library
96 Euston Rd
London...
I think you may be able to get a copy yourself from the BL. Have a look here: http://cadensa.bl.uk/uhtbin/cgisirsi/?ps=quDchwB0No/WORKS-FILE/306020013/123
#10 seems to have Cleverdon's commentary as well as the recording of the poem.
This is where you can order it from...