Do primates other than humans have a consistent sleep schedule?
This is only a partial answer, but a reasonable starting point.

In a blog-article "The Evolution of Primate Sleep", the author identified 3 main patterns - the third pattern seems most relevant to your question.

What he found was that there are 2 divisions of sleep patterns - several primates, such as some species of lemurs are nocturnally active, whereas apes, gorillas etc tend not to be active at night.  Seemingly adapting to the diurnal cycle.  In the article "Mammalian Sleep Dynamics: How Diverse Features Arise from a Common Physiological Framework " (Phillips et al.), primates uniquely undergo monophasic sleep.