The guardian. Whomever is the child's legal guardian makes the appointment.
Mother must be there. Autism, per se, is not typically associated with delays in walking. But, delays of any sort merit attention. Perhaps you could start by offering to go with mother and child for a visit to the pediatrician to discuss developmental milestones and other concerns you have. But the mother is the guardian & will have to be responsible for the child if there are problems, hopefully with your support.
His mother should. If there are concerns regarding any developmental issue, the first stop is at his pediatrician's for a full exam and history (including development). If the pediatrician agrees, the next step is with either a pediatric neurologist or developmental-behavioral specialist for further evaluation and management.