But if we had a nuclear holocaust and only some children of afar off island in the Pacific survived, "time" would cease to exist as none of them would know how old they are, what year it is, or even what day it is. They would only know day and night. Until they invented time and started counting. But the universe would continue in spite of their ignorance and in spite of no one counting time.
Of course time would continue to exist, ... for the islanders would age (as would the rest of the universe), ... which is an effect of time.
Unless, of course, you are also of the viewpoint that a tree falling in the forest doesn't make a sound ... unless someone is there to hear it.
It may be a function of time in a mathematical expression, but not the effect of time.
Age is caused by celluar divisions and cell deaths, by chemical reactions, blastic and clastic phenomena. The 'age' itself is simply a recorded change over a cyclical (repetitive) events (day-night, days. weeks, years...etc.).
The aging and the rearrangement of the universe will continue with or without such a record, but the sound of a falling tree is a physical phenomenon that is absolutely related to the falling tree, whether there is someone to hear it or not.
Everything in the universe is cyclical. Time is simply a count of those reptitions; age is a recorded change in an object compared to those recorded repetitions.
Counting repetitions is an arbitrary activity independent of the cyclical phenomenon. The rotations will continue whether we count the repetitions or not.