To me, the choice of words alters the meaning.

Injured refers to a physical injury, such as a broken leg, or laceration, trauma etc.

Ill feels like a better fit when talking about a condition of pain and suffering that lasts. The speaker wanted to express that the man is having severe health problems as a **consequence** of the accident.

The speaker could've said:

> A man is seriously ill after having been injured by a police car.

You stated that you wouldn't say "ill of an injury". I think what the speaker meant was "ill due to the consequences of an injury".