Giuliani was a moderate Republican but left 16 years ago. He could not be elected mayor of NY today.
Bloomberg was once a Republican who became a Democrat on every issue, save perhaps taxes for Wall Street billionaires. Abortion, 2A, global warming, illegal immigration - he is further left on them than your average red state Democrat.
Bloomberg ran as a Republican because he thought that he probably could not have won the primary as a Dem. He was never a conservative.
That's been true of New York Republicans since at least Teddy Roosevelt. If you're a New Yorker of a certain age, you may remember Nelson Rockefeller, Jacob Javits, Louis Lefkowitz and John Lindsay.