I believe the timer is clockwork.
Touch-sensitive membrane buttons like the ones the use on the front of microwaves are much cheaper than mechanical timers.
Most of appliances provide functions expected from devices of their class and follow obvious patterns that you quickly internalize. I have a running bet with my friends that I can operate any electronic appliance without a manual. The hardest case I've ever had so far was setting the clock on a microwave that provided alternative functionality under "START" button when the oven doors were open. It took me a few minutes to crack that one.
[0] - https://xkcd.com/627/
This is what I miss about older technology - especially dumbphones. The input always worked, and always worked with the same timing. I could operate my Sony Ericsson K800i without looking at the screen because my brain memorized the sequences of buttons and timings of common operations. Joystick press, up, up, press, down, press, wait about a second, press, done. Compare with Android phones which routinely ignore or misinterpret input ("that was a swipe, not a press!"), and hang at unpredictable moments for unpredictable amount of time (usually also ignoring input while hung).
(At least every microwave I ever operated was pretty loud when cooking.)