Is It Safe to Use the Chronograph on a Speedmaster When Slightly Wet?

The pushers have seals that prevent water from entering the case.
If it passed a water tightness test within the last one to two years it isn't a problem.
Wash away to your heart's content.
Maybe even go outside and mow the lawn.
😉

Edit: Per comment from @GrouchoM. Forgot to add that, yes, you can operate the pushers if your hands are wet. If you're at all concerned that you may have too much water in the vicinity of the pushers, just a quick puff of air should be enough to displace it, just like blowing out a birthday candle.

According to other threads (see below), it's ok to use the pushers when wet as long as the seals are in good condition. I guess the only issue there is that you never know when the seals are not in good condition.