Before the lightning hits, you can see the time circuit experiencing bugs and the destination date continually changing between 1955 and 1885 (as you can see in this video). Obviously, we can assume it settled on 1885 since this is when it sent the DeLorean. The date was thus "entered", even if it wasn't intentionally. As to your question, I believe the time circuit, once the time travel has been completed, sets the current time as being the same as the destination time used in the voyage.
A clock doesn't "know" what time it is. It simply calculates how much time has passed since a certain reference point. If you give it a new reference point, such as by changing the current time it displays, it will simply update from that point on. How it calculates time varies greatly depending on the type of clock, but, in 1985, the year the movie was made and originally set, computers and electronical items were already able to keep track of the passage of time through internal clocks.
The DeLorean can safely be assumed to possess one such internal clock and, therefore, keep track of time. When the DeLorean was struck by lightening, the time circuit's clock most likely adjusted it's current time to JAN 01, 1885, 12:00AM after completing the voyage. From then on, it kept updating the current time just like any other clock.