Oddly enough I overthought the first sentence, and imagined the Lord Chancellor was some type of local decorative feature like the Duke of Wellington. Then I realized it's probably just a guy with a fancy title sitting at a table in a pub?
The rest is mostly straightforward to me. The text feels the way it literally reads - a bit muddy?
The streets are so full of fresh mud that they may as well be prehistoric mud flats after a Great Flood. I imagine it's quite a large street leading up a big hill if he could imagine a giant dinosaur making the walk. So I picture basically a solid river of mud rising up in the distance.
If there are normally cobblestones or whatever, they've disappeared beneath the muck. I don't know exactly what a chimney-pot is, but black smoke is pouring from the chimney somethings and mixing with the falling drizzle into dirty soot water. The rain is so blackened - and the weather so dreary - that the city itself could be in mourning.
It's so muddy that the dogs are just dirty shapes in the muck, the horses have mud all the way up to their blinkers... which I read as blinders first, so I imagined it up to their heads and necks, like only the top 10% of the horse is actually visible and most of that is the headgear, and the rest of the horse is mud. I don't know if that's what a horse blinker is though.
The foot traffic feels cramped and irritable in the muck, people holding umbrellas against the dirty rain. It also sounds like a lot - tens of thousands of people walking the same paths. The edge of the sidewalk or whatever at the street corner is probably invisible under the mud, and because of that people keep slipping in the same spots. This pushes the mud more and more in the same directions, forming gross layered piles of muck in specific places against the sidewalk or something, causing more people to slip, adding more to the local mud (compound interest)
The day is so dark and dreary that it may as well be night. Overall, it's muddy, raining, sooty, and depressing. There's a big, wide, muddy street up a hill, filled with a constant flow of unhappy people.
I don't know if I would actually read this for leisure, but I like it. I think I'm on the same page for most of it? But I still have no idea what's up with Lord Chancellor. Is he a person staring out a window at the scene in the street? Does his title imply nobility and fancy clothing? What does the inside of the Lincoln's Inn Hall look like?