Is the snow coming from moisture that has been brought over from Europe (as part of the storm), or is it coming from moisture that was already in the atmosphere above the UK?
Even though only half of the water cycle is explained in my first paragraph, the other half is not needed to answer the question. If you wish to learn more about the water cycle, please visit this link.
So to answer your question, a little bit of precipitation is from the atmosphere, but most of it is from water that has been evaporated from bodies of water nearby. And in the scenario you've mentioned, the moisture most likely came from the Atlantic Ocean.
So the real answer is kinda both :/
The moisture comes from nearby water bodies, and then it's transported to where it's going to precipitate. It is local and transported moisture.
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