In New Hampshire, bodies are sent the Chief Medical Examiner's lab, which I believe is located at Concord Hospital. There the bullet would be extracted from the body, and placed in a bag for transfer to the forensic labs on Hazen Drive in Concord. In the case of a gunshot, I believe that the appropriate person from the lab will be at the autopsy. This is where you start determining angle of fire, so on and so forth.
The question for the forensic lab is whether they have a probable weapon. If they have the weapon, all they need to do is confirm that the bullets match, which is a relatively simple process. Broadly speaking, if the weapon isn't available, the New Hampshire lab can give some parameters on the weapon, age of the alloy used in the bullet, and some other information.
But if the bullet baffles the State lab, it is then sent to the FBI labs for further analysis. At a certain point, if the bullet is in good enough condition, then you can narrow it down to just a few possible weapons that might have fired it.
Question is what is the present level of expertise at the State's lab, and this varies a bit as different people hold down the job.
Definitely the lab person would identify an unusual bullet. What you want to ask the gun experts here is what type of weapon and bullet would be distinct to the time period that you want.
Best of luck,
Jim Clark-Dawe