Assume the time standard clocks and any memories about the time standard are destroyed. Can we recover the time standard again exactly?
Recovering the time standard again means we can determine the date and time that are exactly the same as the destroyed time standard clocks will show (if they are not destroyed).
It depends on your definition of "any memories". If you don't remember what a second is, there is no solution.
If you remember the "old" definition (a day has 24 h on 60 minutes, each of it is 60 s), and live on Earth not to far from now, you can rebuild an approximate time standard. If you remember the modern definition, i.e.
(the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom.)
you can reconstruct it as good as we can now, even if you are in a spaceship far away from Earth.
That would give you a second duration, from which you can define a time-scale. The next-step is to synchronize your new time-scal with the old time-scale. This is basically what historian do when they translate Egyptian or Mayan date in our modern system of datation. To have something precise, you basically need some record of an astronomical event.
The best analogue I can see is the work discussed here and here, where the difference between the solar time and atomic time is extrapolated in the past, before the atomic clock era. There are 10 s error bars around 1700, and 3h error bar for the year 1000 BC.
Imagine for a moment what would happen if, just as a practical joke, someone found a way to stop all atomic clocks, just for a short time. This would cause such a tremendous disturbance in world affairs that the loss of TAI would be a totally insignificant matter! Furthermore, when it came to setting it up again, the phase of TAI could be retrieved to within a few tenths of a microsecond by observations of rapidly rotating pulsars...
-- Claude Audoin & Bernard Guinot, p. 252, sec. 7.3.1 of "The Measurement of Time: Time, Frequency and the Atomic Clock", Cambridge University Press, 2001
Recover the time standard? No. Time is the phase of an oscillator, i.e. keeping track of time is counting oscillations. To recover the phase, you have to have another clock, because a clock is something that keeps track of the phase. No memory = no knowledge of the time.
Frédéric's reply concerns the frequency of the oscillation, which is reproducible. That will tell you the length of a second, but not which second (or fraction thereof) it is.