The normal way to pronounce a date such as "22 August" or "22/8" in British English is "the twenty-second of August."
My question is, do the pronunciations "twenty-two August" and "twenty-two eight" occur as well, and, if so, are they common?
I'll often use phrasing like "twenty-two eight seventy-six" if giving my date of birth over the phone, but that's on the assumption that the person I'm talking to is either reading digits off a form, or typing them into a form, and hence I'm trying to make it easier for them.
In British English we do talk about "9/11", but only because that's what the American media used and the British media copied it. When we had terror attacks later, they were dubbed the "7/7" bombings by the media (they happened on July the 7th), solely in a tie-in with "9/11". This only works for those specific dates as "9/11" and "7/7" are terms in their own right - not because people generally use that format.
In normal conversation it would always be "the twenty-second of August", although "the" and "of" would probably be elided (I think that's the right term).
The short answer to this question is no, we don't use these forms for dates in British English. In fact, many UK English speakers aren't really aware that these forms exists in American English.