The first author pub likely won’t even get published until after the cycle, so just submitting it for review is not worthy of an update. The admins on here will say that anyone can submit something to Nature, but that doesn’t mean it will be published.
There is no way to tell at what phase of review your app is in at each school, but I would imagine that it won’t hurt to go ahead and send the updated info you have. The information will be looked at by the adcom, whether or not they’ve already looked at your app. I say go ahead and go for it.