I'm undecided if Nick is going to come after Harry or not.
I'm certain that Nick hates Harry, but of all the villains we've met, Nick, I think is the most focused on his goals.
Now, if Nick can work into his plans a way to kill Harry, he most certainly will. And based on his stated purpose of acquiring the grail in SG, Harry didn't spoil his plan.
So, I think that while Nick would love to end Harry, I think he'll focus on fulfilling his goals elsewhere from Chicago.