I've never noticed a change because of the weather. However when I wake up and there's rain pattering on the window, it does cover up part of ringing and they're the times where I can get a few minutes lie-in. The same is waking up on a spring morning, with the window open and birds outside are tweeting.
The only thing I've found that fully covers the ringing is when I'm outside and it's very very windy. If I had the money I'd buy a house on a stormy island or a rain forest, would be even better.
Cold weather does do something to the blood that's near the surface, or yeah barometric pressure, so does make sense.