Yes, you may have remedies against the school and/or the bus operator, but the strength of your claim will depend upon the evidence available to establish that the school bus caused the damage to your son's bicycle.
The first priority should be to gather evidence. Since the incident occurred near the school, you should immediately ask the school to preserve and provide CCTV footage covering the relevant date and time. Many schools maintain CCTV cameras at entry and exit points. You should also identify any eyewitnesses, nearby shopkeepers, security guards, or residents who may have seen the incident. Photographs of the damaged bicycle should be taken and preserved.
The fact that you could not note the bus number is not necessarily fatal to your claim if the school can identify which buses were operating in the area at the relevant time. You may submit a written complaint to the school principal requesting an inquiry, preservation of CCTV footage, identification of the concerned bus and driver, and compensation for the damage caused.
If the school refuses to cooperate, you may submit a complaint to the local police station or traffic police, requesting investigation into the incident and preservation of any available CCTV footage. Although the matter primarily concerns property damage, a police complaint creates an official record of the incident.
As regards compensation, if it is established that the school bus caused the damage through negligent driving, you may seek reimbursement of repair costs or replacement value of the bicycle. Often such matters are resolved amicably once liability is established. If necessary, a legal notice can be issued to the school and the bus operator seeking compensation.
Regarding the absence of speed breakers, you may certainly take steps to address the issue. You can submit a written representation to the local Municipal Corporation, Traffic Police, Public Works Department (PWD), or the concerned road authority requesting a traffic safety assessment near the school. In your representation, you may point out the movement of school children, pedestrian traffic, and the recent incident involving the bicycle. Authorities may conduct an inspection and, if found necessary, install speed breakers, rumble strips, warning signs, pedestrian crossings, or other traffic-calming measures.
Therefore, your immediate course of action should be to obtain CCTV footage, document the damage, submit a written complaint to the school, and request identification of the bus involved. Once liability is established, compensation can be sought from the responsible party. Simultaneously, you may approach the local traffic and municipal authorities for installation of appropriate road safety measures near the school to prevent similar incidents in the future.