The nearest I have to an actual answer is here; the new profile pages will appear when the CSS changes go through.
At that link there's a list of SE sites that have been done and a list of sites that have not (including Cross-Validated). The lists are in no particular order and there's no indication of when they'll go through.
There are more sites done than remain to be done.
So my best guess at the expected time is in about 6 to 8 weeks. I think we can reasonably assume that to be the mean of an exponentially distributed random variable, so it doesn't matter when you ask, the answer will be unchanged, even though the most likely time is much sooner than that.