Chosen Solution
Bluetooth won’t connect and clock stops on screen lock
Some hit on your phone recently ? Sounds like a fault of the wifi & BT IC, or on the joints soldering it to the logic board. Quite a common issue on 5Cs as Russ pointed out, but it’s most probably a hardware failure. With a little luck and skills it might get fixed by a reflow of the chip.
Upgrade to IOS 10.3.1 solved the problem on my IPhone5 and my daughter’s Iphone6. Good luck, Russ
