I met with Ann Dinsdale (Collections Manager at the Bronte Parsonage Museum) this week. We met in the library in the museum - a wonderful room with cases full of books and documents - and we chatted about the Brontes and where I might start to investigate their writings on the weather.
She mentioned that letters from Charlotte Bronte to Ellen Nussey would probably reveal many references to the weather - so I've come away with a copy of Selected Letters by Charlotte Bronte - edited by Margaret Smith and published by Oxford University Press - as a starting point.* I can then see if the Museum holds the original letters and arrange to view them.
*There were many letters written by Charlotte Bronte: Margaret Smith has edited a massive three volume The Letters of Charlotte Bronte - which the museum has copies of, but as they are so big and I'm not allowed to take them off site, I'm happy to start with the Selected Letters.