Is the well dressing left in place for a certain time after an initial ceremony, so you can just wander into town during the stated time period and go look at it?

Not long, it would seem:

http://www.historic-uk.com/CultureUK/WellDressing.htm

Thanks, Patrick. The petals would get a lot less lovely after a week, it seems.

Do the townspeople leave them up until they get too ratty looking?

Do we leave London earlier morning in order get to the well-dressing town on its last stated day, only to find that the week-old thing has already been taken down because the flowers got wilted? This is the nub of my question; requiring a crystal ball, though, to answer.

I would guess it depends on the town.