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I would like to calculate the prior probability of active precipitation currently taking place at any random moment during a year.

The area I’m looking at (London, UK) has averages of 601.7mm of rain and 109.6 precipitation days per year (according to https://en.m.wikipedia.org/wiki/Climate_of_London).

I haven’t been able to find any historical data for “time spent raining”, nor for average the distribution of rain intensity (which would also help me estimate this), for any area. Is this data publicly available anywhere?

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Check with the Met Office to see what data is available.

My experience, from elsewhere, is that weather data: temperature, pressure and precipitation are recorded at important weather stations at half hourly intervals, with the data being in the format shown below

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With data in format it is easy to determine the "time spent raining" to a precision of half an hour.

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