You may find this game interesting: https://lichess.org/A9ZRnDcE.
Here Stockfish (black) is programmed to select the move leading to the most negative eval position. White is a human who is aiming to lose (i.e.: be checkmated). White is ultimately successful, which goes to show that really playing to be checkmated is a variant in its own right - we could call it Misère chess - and simply playing negative eval moves, even the most negative eval moves, doesn't make one a good player of Misère chess.
The reason for this is that chess engine positional evaluation assumes the 'optimal' responses (as far as the engine can tell) for subsequent plies. If we were to choose the move most likely to lead to our own checkmate ('selfmate') given both sides on every subsequent move will be doing the same (as opposed to the assumption both sides are playing optimally for checkmate), then we would have a Misère chess engine. I am not sure what the technical challenges involved in developing such would be, but they are likely to be substantial given how far behind the state-of-the-art in selfmate solvers is compared to even the on-the-fly mating capabilities of a standard modern engines.
Lastly to answer your initial question, regarding engines playing the lowest eval moves on both sides assuming current eval metrics, have a look at this paper: http://tom7.org/chess/weak.pdf. The algorithm WorstFish does exactly what you ask. Its results against itself and even slightly less bad players are almost entirely draws. Even a much less bad algorithm like a random mover rarely actually beats WorstFish! So I'm afraid the answer may not be as dramatic as you'd hoped: players just keep offering and rejecting offers of material until it becomes impossible to do so, before swiftly trading down into a drawn endgame.