Agree, I would predict a wet, dull and drizzly Winter. Here in the W Midlands we’ve had very light covering of snow that didn’t last 24 hours for the last ten years. Haven’t built a snowman for more than 20 years.
I’m not sure that the old lore covers a warmer climate.
There is some science behind:
Ash before oak, summer’s a soak.
Oak before Ash summer’s a splash.
Oak leaf bud burst is predominantly related to day length, which is constant.
Ash leaf bud burst is more related to day degrees; the number of days the temperature remains above some datum that I don’t know. This could happen earlier or later in the spring. Basically is saying early warm spring = dry summer.
This won’t help predict winter.