They can get them from the queen, from fleas, from eating mice or other rodents and also from eating birds. The can also get them from dirty litter pans and from mutually grooming each other if they have fleas. You can carry fleas in with you, especially as the weather turns colder because the fleas are attracted to your body warmth. If you volunteer at shelters, or work with rescue groups, be sure you keep an extra set of clothes to change into before going home.
If the parents of the cats have never been wormed before breeding, they can pass the worms to the kittens (actually is more of mom's fault). So the kittens could be "born" with worms - usually roundworms but a vet should verify it.
And if a cat has fleas or even eats fleas when grooming, they can develop tapeworms from the infected fleas.
It's not from being bitten, it's from the cat swallowing the flea when they clean themselves.
From the Centers for Disease Control website:
"By swallowing a flea infected with a tapeworm larvae. A dog or cat may swallow a flea while self-grooming. Once the flea is digested inside the dog or cat, the larval tapeworm is free to develop into an adult tapeworm.
The adult tapeworm is made up of many small segments, called proglottids (pro-GLOT-ids), each about the size of a grain of rice; adult tapeworms may measure 4-28 inches in length. As the tapeworm matures inside the intestines, these segments (proglottids) break off and pass into the stool."
I don't remember the time frame exactly, but I think it can be up to 6 months later. Hopefully someone will be along with a more precise answer for you on the time.