(Note: Using long notation for more simplicity)
I can do it on
White's 3rd move
How:
1. h2-h4 f7-f6 2. Qg2-g4 Qe8-h5 3. Nh4-g6# (White's pieces are bishops, Black's knights.)
Here's a link to it: https://lichess.org/study/6FOSi4Om
Update:
I've been trying for hours now to get it down to Black's second move, and I don't think it can be done. Here's why:
First of all, Black's pieces are knights. It's very hard to checkmate a white king in the center of the board with knights only. I believe you need 5 with an optimal setup (Ke4, Nf6, Nd4, Nf4, Ng4, Ne2). This is... not going to be easy to get at all. And I'm pretty sure it's impossible.
Here's why. It's impossible to mate White on their first two ranks, because literally every possible checking square is guarded. However, if White moves beyond rank 2 on their first move, they'll be in check by one of the black queens.
So, there is literally only one possible square White can be checkmated on on move 3: e3. The only way for White to get to e3 is 1. e2-e4 2. Qe1-e3. However, there's one problem with being mated on e3. Remember the five knights earlier? Well, we've got the same problem. (Actually you need four, because d2 and f2 are occupied.) The problem still stands. You can only get two knights there in time. So, there is no way to checkmate White on move two.