I still live in hope of finding the perfect hat - something that is warm in winter, but shades the face, wards off rain, doesn't blow away, and looks good. And can be shoved in a pocket when not required.
There is no such thing of course. I guess when talking about bush hats you mean the broad brimmed variety. I have family out in Australia and they kindly gave me an Akubra Snowy River - apparently the best selling Akubra in Oz. Made of 'Imperial Quality' rabbit felt. It is very good - my preferred headgear in summer. Surprisingly waterproof. Very comfortable to wear and cool in hot sun. As you might expect, it is none too warm in winter, and can blow away in strong winds. The brim is maybe too broad for the UK.
Lots of people swear by Tilley hats. I like mine, but prefer the Akubra.
In winter I used to wear a deerstalker (not so unusual in these parts) but I found a very good but cheap (£5) hat by Gelert (gear snobs look away now....) - olive waterproof outer, peaked, fold down earflaps, thinsulate lined. Not particularly stylish, but excellent up on the hill.