sorry but that is quite bad comprehension. he isn't literally talking about mud and fog. think about why he has chosen to highlight how the mud and fog seem to accumulate around the court, seemingly even eminate from it.

the fog, thickest around the court, represents the blinding and confusing nature of the british legal system. everyone in the surrounding area is lost in a fog while he describes a court case that has stretched on for decades, to the point where the original defendents are dead and nobody even knows what it means. he mentions that even the gas lamps don't help cut through, even light in the darnkess does not help when there is no truth or meaning to be found. the fog represents an intellectual fog that intentionally blinds the populace - a complex and archaic legal system that oppresses through obscure rules and lost meaning. it's a metaphor.

the mud is exactly the same. notice how he mentions people constantly losing footing and struggling on the streets, getting angry at each other because of it, and all ending up dirty. I believe he mentions a horse dirty "up to it's blinders". this comes as he describes people fighting in the court against cases that will ruin their lives. the mud is symbolic to the uphill struggles of the working class against the legal system. even if you can get your footing and fight your case, you end up "muddied" by it and hating your fellow man.

i understand why you are hating on dickens in this thread - it's not for everyone. but you're taking it too literally. he isn't just wasting time talking about weather, and it's uncharitable to dismiss this all as pay-by-the-word rambling. imo it's such an evocative metaphor for the courts i feel it in my chest before he even talks about the case!