"Change" is another word used to describe coins. Change can also mean to make different or modify. "No change yet"can mean "no difference yet" or "no coins yet". The joke is on the word "change" and how it has different meanings. It also relates to coins which is what the boy swallowed.
A small boy swallowed some coins and was taken to a hospital . When his grandmother telephoned to ask how he was a nurse said ' No change yet ' .