Thanks to all who answered. I've got both watches' alarms working again.

Watch 1, the ~10 year old one, has a metal contact soldered not far from the
reset contact (OK, not a button). It was pressed down and came off when I
pulled it back up. After a touch with the solder iron, preceded by a brief
hunt for the reading glasses, it works fine.

Watch 2, the ~3 year old one, has a little spring as everyone said. It was
still there but was caught on the lip of the hole in which it is seated. I
unstuck it and all is well on that one now. I had to replace the battery in
that one again despite having done so less than a year ago. I don't wear it
much so the light hasn't been on much. It just needs a lot of juice
apparently. I had never replaced the battery on watch 1 so it lasted about
10 years.

Stephen