The "demob suit" included a hat: either a flat-hat or a felt hat. So by (say) 1920, substantially all men in Britain at least owned a hat. So you see photos of great crowds of working men emerging from a factory in the interwar years, all wearing flat-hats.

When I worked in the City in the early 80's, balances between banks were handled by a company called Union Discount, who employed messengers wearing toppers to walk from bank to bank. At that time, some bankers were still wearing bowlers. No other men were wearing hats then. But that is an echo of how it must have been in (say) the sixties. My guess is that hat-wearing declined with the arrival of Brilkreem, quiffs and ducks-arses, in the fifties. These teen haircuts were strictly incompatible with hat-wearing.

A trilby is still worn by many racetrack bookies; wearing a trilby suggests you are a sort of "Del Boy" wheeler-dealer, and not very trustworthy. I think a felt fedora makes you look a bit like a federal agent from the thirties, or perhaps a pushy US newspaper reporter (with his press-pass tucked into the hatband).

For the most part, the headgear I see around here consists of a baseball cap. TBH I barely consider that a hat at all - it's barely better than a knotted handkerchief. And a lot of men seem to wear some kind of beanie.

I am balding, and what fuzz remains is clipped to a #2. I now always wear a fedora out - felt in cold weather, a panama if it's sunny. I was always bare-headed until about 10 years ago (except I was required to wear a peaked cap at school, sometimes).