Gotta be a bit careful here with chronology. Yer man from Cheddar is only 10,000 years old, whereas Neanderthals died out thirty thousand years earlier. It seems that Homo sapiens was the only human species around in Cheddarman's time, as now.
Interestingly, the extinction of the Neanderthals coincided almost exactly with a massive volcanic eruption in Italy, the Campanian Ignimbrite, which left deposits over an area fifteen times bigger than Britain. The volcano was situated at what is now the Campi Flegrei ("fiery fields") caldera, now mostly but not completely under water, a few miles west of Naples. There are still places in the area where there are fumaroles and boiling mud pools, notably in the Solfatara crater at Pozzuoli (been there, got the t-shirt). San Gennaro, the patron saint of Naples, was beheaded there in the third century.   There's a notion afoot among some that the eruption helped to finish off the Neanderthals. There's still a big lump of magma not far under the surface in the area, and Pozzuoli goes up and down like a yo-yo. The town endured massive panic-stricken evacuations in the early 80s when the ground heaved by over a metre in a few months, but nothing came of it - that time!