So to begin with, just because there may be an admission or confession of a crime, police will always continue to investigate to attempt to find evidence that corroborates the admission. This is because, under law, an admission and an admission alone will not be enough for prosecutors to sustain their burden of proving the defendant guilty beyond a reasonable doubt. For instance, in gun possession cases, police will always attempt to find a gun and submit it to their labs for printing and further analysis. This is because if the only evidence prosecutors have regarding a gun charge is a defendant's admission, a defense attorney can try to formulate arguments that either the police are lying about the crime (because no gun was ever found) or that the admission made by a defendant is inadmissible (for any number of reasons) and, without that admission, the DA cannot prove their case. This is why police will always continue to investigate for further evidence, even if there was an admission or confession.
What that means is that police do not have to charge a defendant right away. In fact, the law regarding when charges must be filed by allows police to continue further investigation. This law is called the Statute of Limitations and holds that if charges are not brought within a certain time, charges can't ever be brought for the offense. Depending on the exact nature, a gun possession statute of limitations ranges between 3 and 5 years.
So, just because an admission/confession was made does not mean that charges are (or will be) brought immediately.
Before I proceed with your question regarding due process and being held, I want to pause here and allow you to ask any questions you have on what I've provided thus far.