I've got the latest update for lightroom as of July 2016. 2 weeks ago my I iphone 6s could import photos normally but yesterday it failed. I tried my wife's iphone 6 to see if her phone has the same issue but that worked fine no problems. So it's just the iphone 6s which I'm having problems with. At the bottom left of lightroom import the there is circle dial is spinning. I'll try again later tonight but wondered if other people were experiencing the same issue.
Having tried again iPhone 6s is still not importing. My solution was make a folder and use the mac app image capture. Imported the photos off my phone to this new folder, making sure to check for duplicates and copy to my lightroom folder and this worked fine. I then deleted the folder and checked the imported photos were copied to the correct location.