There are obviously many forced wins on any boards where M and N are at least 8 (including M or N or both infinite), so long as there is a corner of the same colour as the bishop's square.
If the pieces are all in the yellow tinted sub-board and the black king cannot escape the d10-j4-j10 triangle, the position is also won on the full board, because such positions can be (optimally) won on that sub-board without letting the black king escape the triangle. Similar goes for the green sub-board. The same applies to an MxN board.
But the won positions are by no means restricted to such positions. In the position shown, for example, White can mate in at most 33 moves against any Black defence. There is, of course, a significant percentage of similar positions.
There are not necessarily forced wins if M and or N are too small. For example, there are no checkmate positions on a 1xN board.
There are, strictly speaking, also a relatively small number of forced wins on (sufficiently large, i.e. M,N > 2, M+N>6) boards which include no corner of the same colour as the bishop's square but do include a corner of the opposite colour. This includes the 7x7 board with the "wrong" coloured corners you ask about. This is also possible in a "wrong" corner of any board that includes such a corner, e.g. on an 8x8 board.
[FEN "7k/8/7K/8/5N2/8/8/7B w - - 0 1"]
1. Ng6+ Kg8 2. Bd5#**
There are no wins on a board that includes no corners, i.e. where one or both sides extend indefinitely in both directions.
There are drawn positions on any size of board. This is the general case on boards that have no corner of the same colour as the bishop's square and on boards where one or both of M and N are too small and, I believe, on boards where M and N are both large. Here is one example on an 8x8 board.
[FEN "8/8/8/8/3K2k1/8/1B6/7N b - - 0 1"]
Drawn positions are the exception on the standard board (less than 10% of all positions according to the Nalimov EGTB).
But I believe that on a 10x10 board there are also draws by repetition, where the lone king cannot force the capture of a piece, but the side with the pieces also cannot force mate. I think this becomes the general case for large M and N as it obviously is for odd M and N with the "wrong" coloured bishop.
So long as the board contains a corner of the same colour as the bishop's square and M or N remains at 8 or less, but it is not too small, mate will still be forcible generally for finite large values of the other and (somewhat irrelevantly) in as many positions as not for an infinite value of the other.
After reading DanStronger's post, I think my comments on draws by repetition on larger boards are erroneous. These were based on a 45-year-old analysis that I made when I first learned to play the ending (the details of which are now hazy). But I'm inclined to think the analysis was flawed. In that case, the percentage of draws should actually decrease as the board sizes increase.