The official answer is that you cannot promote a pawn to a king, so this situation would never arise. However, this is Anarchy Chess, so let's set that aside.
If this situation did happen, and it is Black's turn, it is not checkmate, because Black's bishop takes the queen. It could not be White's turn, because there is no way to arrive at this board state on Black's turn without one or both kings being in check at the beginning of the turn, and so Black's move would have needed to remove their king from check.
Simply stack the two kings (after declaring "king me," of course). You may now move the stacked kings in any direction. If a piece, including a queen, attempts to capture the stacked kings, the demotion sound from Super Mario Bros. is played, the top king is removed, and the bottom king may capture the attacking piece.
It was widely used up until roughly the 1990s. Then they just came out and admitted to being in a monogamous relationship, as it was more publicly acceptable by then.
weird that you mention that, rather than the actual rule which makes this impossible, that you can't normally promote to a king. clearly OP found an exception to that rule though