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Related: place cells. If a rat is in a room, there are cells that will fire when it's in a specific place. The specifics are pretty cool!

https://en.wikipedia.org/wiki/Place_cell

https://en.wikipedia.org/wiki/Grid_cell

Also related, but not yet confirmed: "Time cells", which are also located in the hippocampus: http://www.nature.com/nrn/journal/v15/n11/abs/nrn3827.html

One can possibly think of the hippocampus as a 'space-time integrator' that continuously indices high-level activation patterns from the cortex and stores them much like a log book. Memories in the hippocampus last minutes to months, after which they are either transferred to long-term memory in the cortex or lost.

One can also think of the hippocampus simply as a working memory - short term content addressable memory. If the rat happens to be working on a spatial task, like in a maze, you get place/grid cells. If it happens to be working on a timing task, time cells. The content is simply task dependent.
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Yes, the HC likely plays a role in working memory for associative, declarative and spatial information. Most of the working memory is believed to reside in the (prefrontal) cortex, though; in patterns of synaptic depletion (less than 15-30 seconds) and based on the 'hot potato' principle in which processes in different modules mutually re-activate previous states to manipulate them.
There must be something keeping track of time in our bodies and brains. I had a neat little lcd alarm clock that I bought as a 13 yr old with my own money, and I had it for almost 30 years. I became so finely tuned to that clock I used to wake up a few seconds before it went off and I would hold the button down. A microsecond blip of the alarm still snuck through but I did this at least once a week for years. Most of the time I would wake a few minutes before. It proved to me long ago that the brain can track time with uncanny precision, given enough conditioning.
In the case of sleep your brain can correlate the time of the alarm to whatever physiological state you are in at that point of sleep, which can create a strong associative effect. You could call it keeping track of time though it sounds more like keeping track of a point in time than keeping track of a duration.

I have the same relationship to my alarm clock.

Interesting. So that would explain why...no matter what time I went to sleep I still woke up at nearly the exact same time every morning.
Paper:

http://elifesciences.org/content/4/e11386

Abstract:

The striatum is an input structure of the basal ganglia implicated in several time-dependent functions including reinforcement learning, decision making, and interval timing. To determine whether striatal ensembles drive subjects' judgments of duration, we manipulated and recorded from striatal neurons in rats performing a duration categorization psychophysical task. We found that the dynamics of striatal neurons predicted duration judgments, and that simultaneously recorded ensembles could judge duration as well as the animal. Furthermore, striatal neurons were necessary for duration judgments, as muscimol infusions produced a specific impairment in animals' duration sensitivity. Lastly, we show that time as encoded by striatal populations ran faster or slower when rats judged a duration as longer or shorter, respectively. These results demonstrate that the speed with which striatal population state changes supports the fundamental ability of animals to judge the passage of time.

This paper says nothing about where the source of the timing signals come from, unfortunately.