The police would find out in the autopsy stage that it wasnt those shots that caused the deadbody dead, but other causes cos it's a deadbody supplied by steve coogan character's friend, wouldnt they?
At the end...Coogan says to give the dead body a good burial site..Which kind of concludes that there wont be an autopsy because the body would have gone "missing"
We have this concept, in the movie-watching community, of "suspension of disbelief". It enables you to enjoy a movie without nit-picking over every other minor detail.
You should try it sometime, it really enhances the experience.
I remember a news story on (I think 60 Minutes) about a woman who was murdered. It was obvious to everyone the LA coroner's office had mixed up her body with someone else. But they wouldn't acknowledge it and the police detective came unglued when asked about it.
They refused to open a murder investigation and it went uninvestigated. The coronor's office got a lot of bad publcity about that time dealing with the OJ Simpson case. They had inadequate resources and a big back log of dead bodies.