They did not have to cover anything up, because there was no proof of a murder or foul play. As Dumbledore says, the killing curse leaves no physical evidence on the body.
Muggle authorities were perplexed. As far as I am aware, they do not know to this day how the Riddles died, for the Avada Kedavra Curse does not usually leave any sign of damage ... the exception sits before me,’ Dumbledore added, with a nod to Harry’s scar.
Half-Blood Prince - page 343
The ministry probably thought that the Muggles would make the wrong assumption, which they did.
The police had never read an odder report: a team of doctors had examined the bodies and had concluded that none of the Riddles had been poisoned, stabbed, shot, strangled, suffocated, or (as far as they could tell) harmed at all. In fact (the report continued, in a tone of unmistakable bewilderment), the Riddles all appeared to be in perfect health -- apart from the fact that they were all dead. The doctors did note (as though determined to find something wrong with the bodies) that each of the Riddles had a look of terror upon his or her face -- but as the frustrated police said, who ever heard of three people being frightened to death? As there was no proof that the Riddles had been murdered at all, the police were forced to let Frank go.
Goblet of Fire page 4
They knew it would be hard for Muggle police to link it in any way to wizards or Magic. There was no proof at all that there had been a murder to cover up.