Didn't the Ottomans declare they were the successors to the Byzantine Roman Empire?What would be the effects had the ottomans declared themselves the new Roman Empire after the fall of Constantinople
They did. So did their predecessors, who called themselves Rūm (Rome).
No one recognised it as legitimate. the HRE viewed itself as the legitimate Roman Empire and whilst other European monarchs may have disagreed, they all agreed a non-Christian power could not be the Roman successor, Russia called itself Rome, and the Greek Orthodoxy would not recognise the Ottoman claim either.