A perfect conductor will completely shield the magnetic field. This is the cause of the famous Meissner effect where we see magnets floating over superconductors.
But ordinary conductors will only partially shield magnetic fields if they are constantly changing magnetic fields.
Thanks for the feedback on my answer:
There are a couple of tangled parts to this question, so I think the confusion lies with making assumptions about the original problem statement. So I am going to make assumptions about those assumptions.
This typical problem of a current carrying wire in a current carrying conduit is analogous to find the electric field with a point charge at the center and a sphere of charge. The point charge has an electric field we get from Coulomb's law, the charged shell contributes a zero field inside (all the fields from the charges on the spherical shell cancel in the interior) but has a field outside that's identical if all the charge was concentrated at the center.
In the problem of a wire inside a hollow cylinder, the magnetic field in region 1 is just the magnetic field (using the usual long straight wire equation) from the wire in the center. The cylinder makes no contribution to the magnetic field in the interior. On the outside, the contribution of the field from the cylinder is also just like a long straight wire. The result is, we just add the two magnetic fields for region 3.
So I think the confusion is about how the cylinder doesn't contribute in the interior. This doesn't mean the conduit is cancelling the field, it just means the magnetic field is just due to the wire in that region. Just like the case of the spherical charged shell.