What is the shortest time period that can be used to find the frequency of a signal burried in noise and harmonics?
if you know that the 5th and 7th harmonic are precise harmonics of the fundamental at frequency $f$, you can use them along with the fundamental to help against the noise.  (the signal is not "buried" in its own harmonics, the harmonics are part of the signal, unlike the noise.)

this is the "pitch detection" problem (even if you're only interested in frequencies around 50 Hz).  perhaps considering computing the Average Squared Difference Function (ASDF)

$$ Q(L) = \frac{1}{M} \sum\limits_{i=0}^{M-1} \left(x[n-i] - x[n-L-i] \right)^2 $$

try computing that for lags $L$ around the period length ($\frac{1}{50}$ second).  pick the $L$ such that $Q(L)$ is minimum.  the noise will keep it from getting to zero, but it will get close to zero.

as to the question in the title of this thread, the shortest time to estimate the frequency is about 1 period ($\frac{1}{50}$ second) plus a little more.