assuming time varies universally, why wouldn't our own clocks be affected so that we would have no way of knowing changes to time?
The "time varies universally" is confusing portion, I assume by that you mean that changes in the speed of time is not some local phenomena but rather it affects the whole universe.
If so then yes, this may be possible physically.
But the problem is would we survive if time slows down or accelerates by say 70% or more?
Maybe we would not because our cardiovascular system may not be able to handle such a change.
If by the "time varies universally" you mean local change in time then it should be noticed somehow.
If we ignore the "time varies universally" then I guess it can't affect clocks because clocks are mechanical and/or digital human made products.
It's hard to say whether changes in speed of real time could affect human made time measuring devices.
For example my hand clock does not have batteries, its mechanical and works as long as I wear it, if I leave it on the table then it will stop working after some time!
Now consider, if the real time slows down drastically so that everything barely moves, and it stays so for long enough time for my clock to stop working (because of the lack of movement) then what would happen?
Time speeds up back to normal but I may notice that my clock suddenly stopped for no reason!
Funny example, but I'm sure similar but more sophisticated technology could be developed to detect change in time on microscopic level