There are lots of guys flying big interesting warbirds in the UK display industry who have PPLs, and turboprop experience is not at all relevant if you are flying behind a big round engine that runs on oil and uses fuel to cool the cylinders!
Yes, agreed. But I believe that a pilot with significant experience on turboprop airliners has an advantage over weekend PPLs, since (1) rightly or wrongly, such experience impresses people, including insurers, and (2) experience of heavy, relatively fast airplanes is generally helpful when it comes to converting to most warbirds.