there is a timecode effect in premiere, it can be set to use the timecode from the source clip, the timecode of the clip as it starts in the timeline, or you can have it make its own time and start at the time you wish. it does count hh:mm:ss:frames so if you dont want the frames you will have to cover it up or hide it, can use a mask for this. you can make a title for the date. for the turnover to the next day, you may have to split the clip on the timeline and reset the timecode and/or date title.
How to make a Digital Clock using the Timecode Effect in Adobe Premiere Pro - YouTube