How much tweaking of figure design (not content) can I postpone until the final figures are requested?

As a referee, if figures are in a rough shape, I will absolutely call the paper out on that, and demand improvements. This will potentially set the acceptance back by weeks. Unless the figures will take weeks to get right—and in this day, that would usually imply that more data needs to be collected—then it's better to just do it right the first time.

I think this will depend highly from the publisher himself, but any problem will probably result in delay before publication.

If the publisher sees for the first time at the final draft slightly new figures, he might still ask for a few modifications. As such, I prefer to do before the next revision as much as possible to make sure that the publisher is happy and everything is done as soon as possible.