This is checkmate, because the checkmated king gets captured first.
One proof can be found in the rules of a chess variant called Grand Chess. It is played on a 10x10 board, and has different pawn promotion rules. A pawn can promote on rank 8 or 9, but doesn't have to, but must promote to move to the 10th rank. However, pawns cannot promote to duplicate pieces (example; two queens, three rooks, or two chancellors (a piece new to this game - it is called the Marshall, but the name Chancellor is more commonly used to identify the rook-knight compound)). From Wikipedia:
" If no captured piece is available for promoting a white pawn about to reach the tenth rank, the pawn must stay on the ninth rank, but it can still give check. This is not as strange as it appears, since pinned pieces can still give check in standard chess."
So that is checkmate, because pinned pieces can give check.
Sunshiny, I like your comment "the king gives the orders." But that actually underscores my point. If the black king captures white's queen, would white king tell white rook "Kill her" knowing that as soon as he does the black queen will kill him? In fact, if we imagine this as all concurrent rather than turn based, by the time the rook moves 5 spaces to kill the king the black queen will already have moved two spaces to kill the other king. So now which King died first??
1. It is turn based. The black queen receives no orders once the king is dead. Or the black king is checkmated aka surrendering and tells his pieces to stand down. Would the black king not surrender and attack the white queen knowing that the queen has back up positioned to kill him?
2. If moves were counted by amount of space crossed, it would throw the whole game of chess out of whack. Instead, you could imagine it as some of the pieces using ranged weapons, and then moving to the spot in order to confirm the kill.