I've never been much of a fan of "mirror" universes, as they're more a literary novelty than anything that might logically exist. In the real world, a small difference in the past will inevitably have huge effects on the present and even greater on the future—the further back that an alternate universe/timeline branches away from your reference universe/timeline, the greater the differences in the present will be.
Something like a mirror universe as portrayed in fiction is pretty much impossible, since these mirror universes are generally caused by some huge historic event playing out differently in the distant past. So you wouldn't have an "evil Archer" or an "evil Whorf", as their parents/grandparents, or even great-great grandparents, etc. would unlikely have ever met or married, as events that originally brought them together wouldn't have occured.
I mean, if being 5 minutes late or 5 minutes early can cause you to never meet your best friend/wife/boss/etc., then what would be the repercussions of sweeping cultural and sociopolitical changes like Earth being an expansionist empire with a militaristic government or Vulcan being conquered by the Terran empire? Surely by the 23rd century the ripple of changes would have become so widespread that it's unlikely the entire Enterprise crew would still exist in the mirror universe, much less all be stationed together like in the prime universe.
So whether the new timeline would have its own mirror universe really just depends on the writers, since the dynamics of mirror universes are completely arbitrary and irrational.
And I'm sure some people will argue that the infinite alternate universe/realities theory means anything is possible, but I personally don't subscribe to that notion. Having infinite realities doesn't mean that all realities must necessarily exist, just as an infinite set doesn't have to include all numbers. (Though it's a perfect way to excuse all plotholes and bad writing. Plus I'd love to be able to imagine that there's a reality out there where Jadzia Dax is my wife.)