How long does it take the body to make a new vein or artery around a blocked one?

Not the way: In most parts of the body, there are redundant elements of the blood supply, so that if one vessel is blocked, the others already present take over. New vessels are not grown to jump over the blockage. This is not true for the brain and back of the eye where blocked vessels create permanent damage as collateral vessel are not present in these tissues.