"Wilfried Menghin, the director of the Berlin Museum, has been extensively studying the hats. According to Menghin, the king-priests “would have been regarded as Lords of Time who had access to a divine knowledge that enabled them to look into the future." - Ancient Origins article
Ancient people may have believed that the wearers of these hats had powers of prophecy because they were able to accurately predict the changing of the seasons.
The oldest of these hats was buried in 1600 BC (this date missing from above article but I found at the actual hat exhibit yesterday in the Martin-Gropius-Bau Archaeological Museum in Berlin) but had been in use for centuries before its burial. One thing in common is that all four of these hats were buried before the introduction of the Hallstatt Culture in these areas.
I am not an expert on Celtic research. I have tried to learn as much as I can about the Hallstatt Culture, have found many conflicting maps, and am not sure which ones are the most definitive.
Going from the map from Wikipedia depicting a core Hallstatt zone along the German-Austrian border, all four Golden Hats are found outside the core Hallstatt zone but within later confirmed Celtic territory.
Did these hats simply "go out of style"? Or were they no longer produced because the tradition was replaced with one from a competing religion? Perhaps the Hallstatt Celts had been practicing a different tradition that did not use golden hats. Where their influence and culture spread, the hats may have been abandoned due to a willingness of peripheral cultures to conform to the practices of the new elite. But the process may have been forceful if the Time Lords were unwilling to simply fade into the dust.
What culture did the Golden Hat people belong to? Given the oldest hat's burial 1600 BC the cult of the Golden Hats predates the Tumulus Culture.
Maybe the Golden Hat people were speaking a Celtic or non-Celtic Indo-European language. If Celtic languages were being spoken in the British Isles at this time, then it makes sense that this area between the Isles and the core Hallstatt zone would also have been speaking a form of Celtic. It may have been the same language as Hallstatt Celtic (later version known as Gallic) or closely related.
Anyone with additional information and links to research articles is encouraged to post.