Wellies will definitely keep your feet dry!
Nearly five years ago, I invested in a pair of Dubarry boots (SUPER comfortable, but expensive! waterproof leather boots) and have worn them non-stop. (Seriously, this year I have worn them for the last 18 months, at least three or four times a week! And I walk a mile and back to work each day!). I love them. My feet are dry (and warm), and they are so comfy. And reasonably smart looking.
There are loads of different brands of wellies though.
I would also second the cheerful umbrella. They make you smile on those days when it feels like months since you have seen the sun!