I know this may be a strange question, is there access to the button itself, as in can you get to the wires of the button? If you can a simple Relay may be you best option.
As for timing, I use these often ( much cheaper ones out there , not sure how much power your project needs to switch ) But you connect up the relay dial in the time you want and set it for repeat, and it will cycle for you.
**broken link removed**
Like I said, if you have the option to get to the wires or pins of the switch it will make life allot easier for you.