In message , Paul Corfield
writes

At its terminus at Hammersmith Bus Station change
to a 391 bus which will take you to Kew Gardens station or else stop
close to Mortlake Road which the records office is situated on.

I'm afraid I wouldn't recommend the 391 for anyone with walking problems
(as intimated by the OP). Getting from a 391 stop to the PRO will
involve a bridge (steps or a long incline) plus a fair old walk. The R68
(from Richmond) provides the nearest access.
--
Paul Terry