I don't know, but 'spoken English' sounds more natural to me. The other does not sound wrong either.
Oral sounds a little more vague, though, since 'oral' just means 'related to the mouth'. If someone says "put a little English on it" in sports, they mean to add add a bit of an extra whip to a pitch or or an action. Basically I'm just saying that, because these words have other meanings, it seems overall clearer to say 'spoken English'.
However, 'oral exam' is a common term, so if someone says 'oral English exam', they really mean an 'oral exam of English', not an 'exam of oral English'. It would be uncommon to hear someone say 'spoken exam'.
I doubt it is much of a Britishism, but I could certainly be wrong about that.