It took me a year with my Walker to get her even remotely use to having my cat around. And she is only okay with the cat if he isn't moving or making a noise. She roughly licks him and smacks her nose into him, trying to get him to run so that she can tree him. But she's never hurt him. She only even tried to pick him up by the scruff of his neck once (she has this fascination in picking up rabbits and other prey by their scruff like a puppy...
Mine never hurts cats but he ALWAYS trees them. He is impossible to change. It's the breed. They are bred to tree animals.
If the cat runs, the Walker will chase and tree it. Our cat has learned not to run. The Walker may jump around the cat and poke him with her nose, but if the cat doesn't run she moves on.
Only if the cat does not run from them, and this is not guaranteed. Most any Coonhound is going to chase, and possibly injure and/or kill a cat. Coonhounds are not vicious dogs, but have a very strong desire to hunt and track. Dogs & Cats traditionally don't mix well, and when the dog in question is a natural hunter, the relationship is even worse. It just depends on how much time and energy YOU want to put into correct & consistent training, and protecting. It will not be something that will happen overnight.
My walker loves my cats and one of my cats are deaf and they love to play with eachother. if they are brought up around them any dog can be good with a cat.
When I got my walker puppy he was 3months old. I already had 4 cats ages 1, 2 4, and 6. They had been around my other lab mix dog before with no problems. Though my walker puppy(he is now 9months) is an inside dog mostly so he likes to chase the cats around the house and we have to correct him. But he doesn't try to hurt them in any way, more like he just wants to play. So I would say with some training and correction they should be fine with cats. Have lots of chewables tous and play things around as a distraction for thenow also so you can redirect their attention.