The Republican controlled Senate will no doubt do everything it can to either prevent the nominee from getting a vote, or defeating the nominee, whomever she or he may be. (And, sure enough, AP just reports that Senate Majority Leader McConnell says Senate should not act on any Obama nomination.)
The President will then have the option of taking the Senate to court over its intransigence, and this suit could well end up in the United States Supreme Court -
Which is now spit, 50/50, between progressives and conservatives. A tie in the Supreme Court would leave nothing changed.
But possibly Justice Kennedy or Roberts would do the right thing, and strike down the Senate's failure to act on the President's appointment.
Interesting times ahead!
And here's a video of Hillary Clinton letting the Republicans know what the Constitution is about...