A couple of comments:
- I don't really see Ottomans developing in this TL; they were the result of being placed on the Byzantine frontier where they were able to capitalize on the campaigns against the dying empire, absorbing traditions and ideas from Byzantium, Persia, the Arab world, and Central Asia. In this TL, they are likely to be just another tribal group in the Mongol hordes.
You could argue that the founders were extremely talented and thus establish them somewhere, but they will not be OUR Ottomans.
- It doesn't seem likely to me that the Seljuks will survive this; they are too potent for the Mongols to leave intact in such a critical location. Perhaps a Mongol thrust into Asia Minor, plus Western preoccupation with the Mongol invasion, might actually allow a Byzantine revival! If anyone has the diplomatic chops to talk the Mongols away from them, it's the Byzantines, who don't really have anything they want anyway. Without pressure from Venice, Bulgaria, Serbia, and the Seljuks/Ottomans, there is really nothing to prevent the Byzantines from regaining lost ground. As allies of the Mongols they would actually be quite useful, and you could even see some Orthodox Khanates.