I've recently been experimenting with the Arduino microprocessor boards and have found them remarkably easy to work with.
I grabbed one from ebay for about $10AU along with a servo motor for less than $5AU. All software and coding was open source - free and easily accessible - communication is via a PC, once it is programmed you can set and forget. Altering the program would be quick and easy.
The arduino runs on 5V (a USB cable) so sourcing a power supply would not be difficult if you have a spare USB phone charger. The servo motors generally come with shaft attachments and all you would need to do is glue something the correct length onto the servo shaft attachment that would hit your appliance button. You would also need to fix the servo in place, if you didn't want to screw into the appliance you could construct a frame that sits under the appliance and holds the servo in place.
Alternatively, as someone else mentioned, an electromagnetic solenoid (a small unit that pokes a pin out when energised) could be simpler in terms of coding.
The arduino would need to be constantly powered to count time as it has no battery to power an internal clock. There is external clock modules that can be bought to provide for this.