Joes:
What do you want to happen when the end of the 5-to-25-minute time interval is over?
nothing just want it to variate an output over a period of time when a switch go's high
"Variate" isn't a real word. Do you mean "vary"? If so, how do you want it to vary - just a linear change in the duty cycle starting at zero and ending at the maximum defined by your pot? Is it possible that either of the pots will be adjusted during this timed processing? You mentioned a switch - do you need to deal with the switch going low or high after it has triggered this process? Do you want to respond to any further changes to the switch state during the time processing? How about after it has completed?