Definitely a warm pair of waterproof boots, gloves, scarf, coat preferably with a hood.....if I'm going to be outside for a long period I wear a toboggan also.
You'll need to keep warm but much will depend on the size of your dog and how fast you walk it.
I once saw a woman, walking a Great Dane on a leash, run through the traffic in a big city. The dog loved it so much he raced up an alley dragging the woman behind as she screamed, "Slow down you mongrel!" That was back in the 1970s but it's a picture I'll never forget.
Get boots a size too big and double up on socks, or get silk ski socks. Gloves with "leather" or other non-cloth palm and fingertips so you can manage poop bags. Thermal t-shirts under your jacket/coat. Hat and scarf.