How long does it take for potato dough to become thinner?
If your potatoes are prepared correctly before adding the other ingredients, then excess water should not be a problem.  After boiling your potatoes until tender and draining them, I recommend placing them on a sheet pan and drying them in a 300° F oven for 5 minutes or so before passing them through your food mill or potato ricer.

With properly prepared potatoes there should never be a problem with thinning.  In fact, prepared gnocchi can be covered with plastic wrap and refrigerated until ready to boil without fear of disintegration.  With most fresh pastas and dumplings, time delays in preparation usually pose the greater threat of allowing the dough to dry out.

The secret to delicate gnocchi is to use just enough flour for the dough to come together and to knead that dough only briefly.  More flour and more kneading generates more gluten and that means tougher gnocchi.