This may sound crazy but it is inspired by What if a spacecraft lands on my property?
The answer there seems to be that you don't acquire another's property just by it appearing in your domain.
The conclusion I draw from this is that if someone shot you, they would legally be entitled to ask for return of the bullet.
If so, and return of the bullet were refused, could this be raised as a civil case and how would one assess the damages?
It could probably be argued that a bullet intentionally shot from a gun is abandoned property, and thus the shooter has no claim to its return. In addition, such a bullet might be evidence of a crime, to be seized by law enforcement, although that would not affect its ultimate ownership, at least not in the US.
Intent matters in such cases. A person who shoots a gun probably does not reasonably expect to retain possession and control of the bullet, whatever may have been shot at. Well, someone shooting at a properly controlled target for practice on his or her own property probably retains ownership of the bullet.
As others have alluded to, there are a number of different scenarios that can be read in to the OP. Here's one...
Re: the United Kingdom
If this is a criminal act, and if the bullet can be recovered and if the police are investigating - the bullet will be seized as evidence and retained as police property.
Once any investigations, trials and appeals have concluded the police will either apply to the court for an Order of destruction or (if it is unique, notable or has a significant importance) an Order for it to be permanently retained by the police/army etc to be used in training, for display in a museum etc.
Even if found not guilty, it may not be returned to the shooter unless he makes a successful application to the court contesting the Order.
(NB this reply does not describe the process for every scenario involving seized property, but it is the norm for weapons etc that have inflicted injury.)