You could compute a median of means, where you count clusters as single points by taking the mean of clusters as a single point.

You could compute the emperical distribution function along with a confidence interval. In your case you have a detail which is that your measured speed $V_i$ has some error $\sigma_i$ relative to the actual speed $U_i$, and you want to know the median of $U_i$. You could use a Monte Carlo approach to estimate the error in the estimated median.

Possibly simpler would be to compute mean of means which allows an easier estimation of the error.

Here are three possible approaches:

1. You could compute a median of means, counting clusters as single points by taking the mean of clusters as a single point.

2. You could compute the empirical distribution function along with a confidence interval. Since your measurements have errors, you could also estimate the error in the estimated median with a Monte Carlo approach.

3. (Possibly simpler) You could compute a mean of means, which allows an easier estimation of the error.