One of the Discworld books, I think it was Lords and Ladies, involves one of the characters being able to see into other multiverses of her life, and there is literally one for every single different decision she could have ever made. I agree with Calista's idea that it would just create a completely different universe of the combination of the two mirrors, but the fact that a new universe is being created might cause some issues with how aware the characters are in the original universes (and any others that might exist) of the combination.

Theoretically, this merged universe could already exist somewhere along the timeline of the Prime Multiverse and the Mirror Multiverse, and its own timeline could have started just as the combination took place. There's also the issue of the people in THAT universe knowing or not knowing about their being from two merged universes...you'd definitely have to do a lot of explaining on those parts.

I don't think it would work well for an entire IAHF to be replaced by another one, because then you could get into the issue of where the replaced one would go. After all, OFUs don't just disappear. Would it get absorbed? But then you get back to the idea of replacement...maybe it would go into the void, that space between universes like in Doctor Who? There's a lot involved you'd have to think about.

I hope the advice helps. Discworld novels do have a lot to do with different dimensions and universes being traversed or seen into or talked about, so maybe you could read a few of those (but there's a lot, so you'd have to have a lot of time on your hands to read them all). Good luck with the story!

I think it would create a third multiverse which is a combination of the first two.

A story is not just a single storyline; it's also the combination of all the possibilities that could have happened. Therefore, because you are writing a story about sets of storylines, your story is the combination of all the possible ways those storylines could combine. In this case, that means that both the combination and the separate state exist after the merger.

Yeah, this is how you get thirteen dimensions in string theory. :P