The covering for one's bonce, in my young days, at least, were known as:

me tile
me 'eadgear
me titfer (tit for tat + hat)

On a more formal note, one referred to:
a Trilby (soft felt hat)
A Homburg (VERY formal hat with a stiff brim)
Cap
A Bowler hat (what you USns would call a Derby)
A Top Hat (VERY, VERY Formal - for State Occasions, etc)