As for AE's ideas, remember that Spain forbade it strictly any Muslim presence in their American colonies. Unless again the fate "guide" them to, Spain would never let any Mudajeras set their feet on America.
True enough. That would have to be changed somehow.
But I can't see the Ottomans (or the Byzantines, or anyone else in that region) ever getting American colonies. Why would they bother? The main reasons for European colonisation of the Americas in OTL were
1. (1492-1500s) We've found a westward route to Japan and the East Indies!!!111
2. (1500s-1700s) Oh...wait, no we haven't. Dammit. But, maybe this continent is narrow enough for us to march across it and sail from the other side to reach Japan and the East Indies.
3. (post-1700s) No it isn't. Bugger.
The whole reason the Europeans were trying to get to Japan and the East Indies in the first place was precisely
because the Ottomans held a stranglehold over many of the eastward trade routes, especially the land ones. What reason would the Ottomans have for wanting to plant a colony in America? Besides the fact that it would be even more far-flung for them than it would be for the European powers in OTL.
You could get Muslim colonial powers in America, of course (Mali before 1400 or so, perhaps Morocco later on) but Constantinople, I think, would not be one of them.