That's a really good question you've asked. How long you have to search for the owner before you can call the dog your own, would depend on the local laws re dogs. The dog refuges & shelters in your area would have to obey these laws so they would know how long they have to hold a dog before offering it for rehoming. So I'd ask the question of the best managed dog refuge in your area. It would be a good idea to speak to these local dog refuges about your found puppy in case the original owners have been contacting them to see if their puppy has been handed in . But tell the refuge you're happy to continue fostering the puppy until it can become your own...when the required time has passed.
What good people you & your mom are to take such care of this puppy. I hope you keep telling us all about it...like what name you give it (if you get to keep it) & what it looks like & what it likes to do.