If you already estimate the empirical cumulative distribution function ,
or equivalently the empirical survival function ,
The asymptotic approximate standard error of the estimate is available
(at least for the common Kaplan-Meier case)
But if you have only interval censored data, without any observed failures,
traditionally you do not have a unique maximum likelihood nonparametric
estimates. You may search for the Turnbull's interval/algorithm as well.
(I am not an expert in survival analysis)