'Bernhard' with an H is German; in English he's St. Bernard.https://en.wikipedia.org/wiki/Saint_Bernard_(...https://en.wikipedia.org/wiki/Bernard
You're right to notice an AE/BE difference in the pronunciation of that name. But is it different for the dog (or the saint of Clairvaux) as opposed to the playwright? I couldn't begin to speak for BE, but Wikipedia gives one pronunciation for 'St Bernard' that clearly isn't AE, so is it BE?
St Bernard BE /sn ˈbɜː(r)nə(r)d/ (???)
St. Bernard AE /seint bərˈnɑːrd/
I feel more secure about the playwright, who follows the usual pronunciation on either side of the pond.
George Bernard Shaw BE /ˈbə(r)nə(r)d/
George Bernard Shaw AE /bərˈnɑːrd/
Sorry, that doesn't really address your question about 'St.' But you already seem to have noticed that in BE it may have no vowel at all, whereas in AE it rhymes with 'faint' (bzw. 'feint').