Scientifically speaking, time can exist even when we are not around to measure it. Yes, we invented the scale in which we measure time, but we can assume that this measure extends into the past as far as the beginning of the universe just as we assume it can extend into the future.
As for how we measure it: we use the standard model of cosmology, the big bang theory, and measure the recession of other galaxies from ourselves which gives us Hubble's law (or something close to it, with higher order corrections for the period of "inflation"). From this, we can then determine, roughly, how long the universe has been in existence. Of course, it'll not be exact, but we reckon it's something around 15 billion years.


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