Maybe an easy coding modification? After 15-30 seconds, the button starts flashing. After a minute, a soft series of, say, 2-5 beeps.
These could be options that could be turned off (if, for example, a player has a mobility issue and will always be taking an extended time. But I presume that, in those cases, such a player types out this information immediately, so that the other player knows going in that the game will last 2 or 3 or 5 times longer).