The ones that can.
Your viable dog population is going to be a fraction of the total.
If the humans leave their dogs behind in the cities, they are (mostly) not going to leave them locked up indoors or in yards. They'll probably cut open their food bags and fill some water bowls. Which gives the dogs a week or two to figure out other food/water sources and social structures.
Dogs who can reproduce will be a small portion of all dogs. So they will compete for food and other resources with the dogs who can't reproduce.
Most pets are fixed (sterilized). Many ferals are fixed (while captured feral dogs are often fixed then put up for adoption, feral cats are more likely to be fixed then released to keep the population stable...some feral dogs will actually be strays, not true ferals, and others will be released ferals. This varies a lot country to country and, even within my country, the United States, it varies by community.
Large portions of dogs who are not sterilized will be: too young, too old, or in puppy mills (which render mothers incapable of surviving on their own). Some of the puppies may survive if protected by other dogs.
I'm taking you at your word that the "general ecosystem" hasn't changed (though for some reason all the humans leave?) so dogs can form packs and hunt small to medium animals, drink fresh water, and not die from radiation or bio-weapons, etc. Some dogs will have access to houses (with doggie doors or other openings) which protect from the elements and help protect from predators.
Which breeds will do the best?
Small to medium-sized dogs who are suited for the climates they live in.
Very tiny dogs, teacup dogs, say under 6 lbs or so, may find niches if they can eat mice and so forth, but mostly will not do well. They won't be able to fight off larger dogs for access to food. They are even smaller than most cats but with less fighting power and the inability to leap or climb like cats can.
- Toy terriers
- Toy poodles
Small dogs, in the 6-15 lbs range, are more cat-sized and can eat gophers, mice, maybe rats (most cats avoid rats), birds, and insects. If they can avoid predators, they should do okay.
- Chihuahuas (the larger ones)
- Smaller Terriers (various breeds; they're good ratters)
- Miniature Schnauzers
Medium dogs, say about 15-40 lbs, can eat rats, gophers, badgers, squirrels, birds, etc. They have a decent chance of finding food.
- Larger terriers
- Dachshund (some hunts animals like badgers)
- Miniature American Shepherds
- Miniature Bull Terriers
Individual varieties within each breed will have different weight ranges and of course any individual dog may weigh more or less. So don't take these listings as definitive.
Also, you didn't specific location, you just said "cities." Dog breeds that tend to do well in Cairo will die in Anchorage. And vice versa. Even if you limit it to climates without super harsh winters or summers, in areas with natural water and urban wildlife, there's still going to be differences in which breeds do better.
What about larger dogs?
Larger dogs, say 40 lbs and up, will probably do better in the short term. There will be a lot of competition for food and access to water and shelter. Bigger dogs will generally win over smaller dogs.
Dogs in the 40-80 lb range might end up okay, but much larger dogs will not. Why? Because they require too much food. Unless the urban areas are small and next to wildlife areas, the large dogs will run out of food within a generation or two, if not sooner. They won't be able to breed and sustain their young. Some larger dogs will leave the urban areas, but that wasn't part of your premise, so we'll ignore them.
The actual breed won't matter so much.
Most of the dogs adapted to the climate who are able to breed will be strays and ferals and mutts used to living outdoors. While many purebreds are left unsterilized, they may be inbred, not bred for local conditions, or indoor dogs. Of course you'll have some working dogs who are purebred or close, but not as many in cities as in the country (or even the suburbs). City dogs are generally pets and usually chosen to do well in smaller homes.