Is there any open source, stochastic rain forecast data available for the UK?
Well there is ensemble data that you can use. For the UK, you can probably use the ECMWF ensemble prediction system data or GEFS from the US NWS. Both are global models, so they can be extended beyond their respective centers.
These ensembles can be used as samples of the probability density function. Normally (pun unintended), with these types of ensembles, the pdf can be assumed to take the form of a Gaussian distribution per ensemble filters. Obviously, this is problematic, as it implies a probability that there is negative precipitation, which is unrealistic. However, if you want to be clever, you can probably fit a kernel to the data. Or you could choose your own distribution. The distribution that is most widely used is the Gamma distribution, though the Pearson distribution has been shown to perform well too. Perhaps using the Kolmogorov-Smirnov test could help you determine which distribution is best.