That...has some interesting implications. I agree that the parallel universe will likely mess with Harry's mantled state. Maybe while over there, like you say, he'll be cut from Winter and unable to access its benefits. It might even cripple him again. But unless something really weird happens, I think the agreement he had with Prime Mab (which would be an absolutely terrifying transformer) would not be broken just be moving from one universe to another.
For what it's worth, I don't think that Harry's back is explicitly tied to his gig as the Winter Knight. I think the bout of paralysis he experienced in Cold Days was a result of him (briefly) reneging on his Deal with Mab.
The actual text of Harry's Deal in Changes reads (Changes, page 245 on Nook):
"You know my price, mortal. Speak it."
"You want me to become the Winter Knight," I whispered.
A laugh, both merry and cold, bubbled beneath her response. "Yes."
"I will," I said. "With a condition."
"Speak it."
"That before my service begins, you restore my body to health. That you grant me time enough to rescue my daughter and take her to safety, and strength and knowledge enough to succeed. And you give me your word that you will never command me to lift my hand against those I love."
Emphasis added.
First, obviously, Harry's paralysis is taken care of before his term as Winter Knight starts, and that service (as I understand it) begins before he has the Mantle; killing Lloyd Slate is actually the first command she gives him as Queen to Knight. Second, he's agreeing to service explicitly in exchange for healing him (and, you know, buffing him up with Winter Juice).
All of that is to say that by saying "Screw Winter Law," he tacitly rejected the terms of his Deal, which yanked his non-paralysis away temporarily.