The nomination starts exactly when the new GotW is selected, which is done by a script at a set time. You can see the exact time by hovering over the time in "will be selected X days from now" on the GotW page. Currently it's set to 08:00 (am) UTC on Sunday, which is a recent change. It was 17:00 UTC on Sundays for a long time.
The nomination ends after 15 choices have been submitted, and then voting begins. I think this usually happens within at most a few hours after nomination has begun.
I like the idea of switching to nomination lasting a set amount of time, rather than until a set number of choices has been reached. This would run into issues if GWM happens to have a longer outage on the nomination day, though. An alternative would be to increase the number of choices required until nomination ends.
I have been trying to nominate a candidate for several weeks, but have not been able to so far. Was the time for nominating candidates shortened? The time available is not clear at all. Is there a maximum number of candidates?
Suggestions: