First of all, Sharpe Ratio (SR) is meant to assess the uncertainty surrounding the expected returns of your PnL. In short: you divide by the standard deviation of the returns because you trust less a time series of PnL with a large standard deviation than with a small one.
Nevertheless it is in fact not the best indicator; the best one is the t-test, that reflects the probability that your PnL is positive, under Gaussian assumptions. There is a simple relationship between the SR and the t-test: when your returns are annualized and known over $y$ years, one can write:
It is natural since the longer you observed a SR, the more reliable your expected returns.
It seems that you have in mind to compute different estimates of your SR over different time windows. I would say that it is in the spirit of bootstrapping the SR. My view is that is is better to bootstrap the returns and to directly compute the probability that the returns are larger than a given threshold. You can do it as soon as you compute the returns $r$ over $N$ different time windows $\omega_1,\ldots,\omega_N$. Hence an estimator of the probability that the expected returns are larger than an arbitrary threshold $\theta$ is the number of $r(\omega_n) > \theta$ divided by $N$.
Then you ask the question of the time scale at which it is "better" to compute the returns and obtain a reliable SR. If the returns at the smallest time scale are i.i.d., then all time scale will give the same estimate, and hence it is better to take the smallest time scale (to use as much points as possible).
If they are not i.i.d. it is far more complicated. See The Statistics of Sharpe Ratios, by Andrew Lo. It is for instance obvious that is the returns are mean reverting, the largest the time scale, the lower the standard deviation and as a consequence the largest the SR.