Awesome project! Reminds me of donothingfor2minutes.com from Calm, but with a different end goal of focus instead of calm.
Regarding mobile phones going to sleep, Wake lock [1] might help, unless you can reduce to 59s since I believe 1m is the threshold (make sure to request within the context of the user hitting "start"). Unfortunately on older mobile browsers [2], the best workaround I found was using this NoSleep library[3].