You'll be fine in the sense that you won't be checked out of hotel 1 when you check into hotel 2. You can be checked into both hotels at once.

But you won't get Honors points or stay or nights credit for the one night stay in the middle. You'll have to email or call in to get that and explain what happened. In my case, I have checked into one hotel with my wife and then have had to go to another city for work, with my wife staying in the original city for the rest of the planned stay. Once I wrote in and explained (to [email protected]), I got the points for the stay in the second city. I've never asked for additional stay or night credits, just the points.