Priority baggage isn't an advertised benefit of status or cabin on BA but is across the rest of the major OW carriers. The tag is added at MAN or whichever other airport I'm at through the F/J checkin (even though travelling Y) and it doesn't make any difference to whether it comes out first or not. As far as I was aware, this was a blanket thing for anybody who is eligible to use the F/J checkin desk.

On US Airways recently, the same happened but the bag came out in the priority batch at both ends.