AMC own a stake in BBC America, which is not the BBC itself, but a commercial operation part owned by the BBC.

As the BBC is basically owned and paid for by the British public, no part of it can be sold into private hands.

Yes, HBO Max have the rights to all 12 series. The deal covers Seasons 13 through 15, but the show will still premiere on BBC America. So really MAX only has exclusive rights to re-runs.