OK, so this is an interesting question. To start, I'm a circadian researcher, specifically focusing on how bacteria can influence our circadian clocks.
It is indeed correct that most animals utilize the sky (specifically, the ambient brightness) to determine the time. But the circadian clock is incredibly entrenched. It evolved ages ago, and so by extension, virtually every single animal inherited the same circadian clock (with some modifications). Animals as distinct as fruit flies, fish, and humans have similar circadian clocks. And the circadian clock is unbelievably important, more than people give it credit for. Night and day are incredibly different environments, and every single animal needs to be able to predict and accommodate for the cold that comes at night and the UV radiation that comes during the day. And there's a plethora of other, subtle changes to the environment that we don't fully understand yet. For instance (and probably a bit unsurprising in hindsight), the population of bacteria in the air changes at day versus at night. Soil bacteria, for some reason, act differently at day versus at night. Presumably plants (which are definitely circadian) are influencing the soil bacteria in some way through their roots, but it's not entirely clear.
An interesting consequence of the importance of the circadian clock is that animals have evolved multiple, redundant ways of telling time. If you lock a person in a dark box for weeks to months (scientists have performed this experiment in the past), the circadian clock running within the person is still able to somewhat-accurately tell the time, and we can use experiments like this to tease apart how the circadian clock utilize different cues to figure out the right time. One of the more interesting cues (and the one that I research) is how our bodies use surrounding bacteria to tell the time. And it's known that eating food affects the circadian clock. Food availability, of course, is pretty circadian, especially if you eat food that is circadian. If a mouse comes out at night, then there's no point hunting for mice in the day. I'm not too knowledgeable about deep sea animals (I'm really more focused on mammals), but a quick literature search suggests that deep sea animals do have circadian clocks, inherited from the same ancient ancestors that we got ours from. The conclusions appear to be similar to what I've said above - namely, that even if there's no sky, having a functional circadian clock is necessary simply because other things, such as food, are themselves circadian.
So my answer is, presumably, that deep sea animals can already accurately tell the time, presumably by keeping track of when they last ate