There is a lot of info missing, so this answer will have to be pretty general.
Legally, the statute of limitations controls. You didn't indicate what kind of crime (and I'm not asking you to), but the statute of limitations will vary depending on the level of the offense.
Police and prosecutors are busy, and often will move to get things off their desk, otherwise a backlog can quickly develop. Thus, they are likely to move on a case soon, or not at all.
HOWEVER, if the case is significant (murder, rape, etc.), or particularly complicated (financial crimes can be this way), then investigations can take a longer time.
If you are under investigation, the time to consult with an attorney is NOW, not after you are arrested.
This answer is provided for informational purposes only. It does not constitute legal advice, nor does it establish an attorney-client relationship. Speak to an attorney in your area to protect your legal rights.
Other than the statute of limitations there are no definite time limits on when to start or stop an investigation and file charges.
David R. Damore, Esq., Damore, Delgado, Romanik & Rawlins, Daytona Beach, FL, (386) 255-1400. Mr. Damore is a former Volusia and Broward County Prosecutor and an experienced criminal defense lawyer handling cases in both State and Federal Courts throughout Florida. The answer provided is for educational purposes only. It does not constitute legal advice or create an attorney-client relationship. You should consult an attorney for legal advice regarding the facts of your specific case and designed to help you with your personal needs.
It depends on the crime that was reported. If it was a minor offense, it's likely no charges are forthcoming. If it was a serious offense, the prosecutor has plenty of time to investigate before filing charges. For a 3rd degree felony, the statute of limitations starts at three years. The more serious the crime, generally longer they have to prosecute.