It's completely up to CPS. There are no maximums on how long they can investigate.
The information is for general information purposes only. Nothing stated above should be taken as legal advice for any individual case or situation.
Despite being no specific time frame, the department attempts to initially investigate and then come to a conclusion. The conclusion is a letter that states founded or unfounded. Typically, you might get a letter between the end of the investigation and 90 days after it ends. Since you do not know when the investigation ends - hard to predict the latter timeline. You can make a public disclosure request for CPS records of your own kid. Things like that force people to pick up a file. No harm in making that request while you are waiting.