How do you estimate the error on the height/width of a Gaussian?
If you know the number of overlapping Gauss functions, you may write them as $\sum_{i=1}^n a_ie^{-(\frac{x-\mu_i}{\sigma_i})^2}$, then subtract your time series, with more than $3i$ items, to reduce the risk of an ill-posed problem.
Then apply a root mean square (RMS) minimization / least squares method to infer the $3i$ parameters.
If your FWHM, or $\sigma$ is known to be the same for all peaks, your number of parameters reduces to $2i+1$.

See also Rietveld refinement as a related approach.