I will not start this post by saying "With all due respect to Roger"!!
However, the pawn can be promoted to any piece and it is mate provided the promotion is made before the fallen flag is noticed. The Rook on g8 is protected by the bishop on a2 so promotion to Q or R is not the only way of getting mate.
If the pawn on e7 has been touched then the only legal move is e8 = X mate. Although this move is not complete (or even made) it does mean that Black cannot get checkmate by any means and so the game would be a draw.
Remove the Bishop from a2 and you have a more complicated situation. Even moving the pawn to e8 is no longer sufficient to get the draw as it could be promoted to a N or B. However, if White has not made a previous illegal move, e8 and pressing the clock before flag fall is a draw at least. Under the Laws the pawn must be promoted to a queen and so Black cannot get checkmate. It is then a question of can White queen the pawn before flag fall and get mate. If it is a second illegal move then it would be a draw.
Similarly (with B removed), if White pressed the clock before it was noticed that the flag had fallen, even though it is obvious that the flag must have fallen before the clock press because the flag is only considered to have fallen when it is noticed, the game would be drawn.
The flag can be used to determine which clock ran out of time first when a win on time is being claimed.
Simple really!!!