How many years pass after a death for it to go from a police investigation to archaeological evidence?

Not an archaeologist here, but I do study and practice criminal law on the law enforcement side.

I can tell you that in modern western (US/UK et al) legal systems, there are limitations on the duration of criminal investigations into most crimes. These are written into the criminal law, and are usually called a Statute of Limitations.

There are exceptions to this, and the most common is murder. If there is evidence of 1st degree murder, every jurisdiction I can think of has no statute of limitations. This means that the case remains open indefinitely until solved.

This stays more or less true. Certain famous unsolved murders typically remain officially open for as long as there is still a file on the case. My understanding of the Whitechapel murders (the Ripper cases) are that they are still officially open, though there is no active investigation.

That's the key point to my answer. At some point, though cases remain open, investigation stops. A large police force might investigate any new information that comes up, but they stop looking for new leads themselves. The point at which this happens varies wildly and depends on how big of a case it is, how big the department is, and how much information they had to begin with.

There is also the practical consideration of prosecution. Once a certain amount of time has passed, it is no longer possible for the perpetrator to realistically still be alive. No modern jurisdiction would waste resources proving up criminal charges on a dead person, there is a legal doctrine called abatement that allows prosecution to waive the case due to death. It's a way for the State to say that they aren't inclined to actually drop the case, but they don't want to waste time prosecuting a dead person. In that situation, the case remains open in some jurisdictions, while others will simply close them administratively in order to keep their file system tidy. These administrative closures can be reopened and reclosed essentially at will, if new evidence ever comes to light.

To sum up, half the answer to your question is that modern death cases stay open until solved. At some point this becomes a technicality, and that point moves based on the particulars of the case. Eventually, it becomes archaeology but that point also probably moves, and I have no idea how to begin defining what is or isn't archaeology.