Don't know if you've sorted this yet, but I've been using my F50 for a while now.
Only just got round to checking the video (as it would always say it was recording I hadn't worried about it too much).
Of course looking at it on a PC it appeared that I'd not recorded anything since October 2016!!
Thinking that it served me right for not checking, I reformatted the card and tried again.
Checking again I notice the date was still showing October even though the recordings were for that day.
Resetting the time/date in the app didnt make any difference.
I'd upgraded to Kingston 64Gb SDXC Class 10, so went back to the supplied 8Gb SD.
Of course it worked and immediately recorded the correct date/time, which was odd as it had sat in a drawer for some time and would have had the wrong date on it.
Seems to suggest that the dashcam knew the time but was unable to write it to the card.
So try another card.