Barbara -- Although sometimes the approach of write it and then figure out how the real world operates works, it probably won't in this case.
Very broadly speaking, when the child is abandoned, a petition for abuse/neglect would be filed by child services. There would be several hearings, where two things would happen. The child would progress from temporary placement, to long-term placement, to foster care. How quickly this happens would depend upon the child, its age, and how the state has this structured. Meanwhile, the state would be attempting to contact the mother, as the court determines whether the mother (and father) had abused or neglected the child, and whether the rights of the parents (both) need to be terminated.
Once the court determines that the parents' rights should be terminated, an entirely new process, or phase of the process, depending upon how the state structures this, to terminate those rights begins. Even more strict requirements to notify the parents exist, so even stronger efforts to contact the parents will be made.
Once the parental rights are terminated, then the child is available for adoption. Once the child is placed for adoption, an adoption proceeding commences. At this point, there is less of a requirement to notify the natural parents, as their rights have been terminated. But making sure the adoption is the right step can take some time.
Time span for this can be a couple of months to years. Failure to show usually results in something along the middle of the time span. Steps can be combined, based upon the facts of the case, agreements reached by the parties, and state law. But notice how even in this very simplified explanation how complex everything is. It's not something that you can wing without having an idea of how the system works.
Jim Clark-Dawe