Based on the Ministers for Magic article on Pottermore I'd have to say No.
The Muggle Prime Minister has no part in appointing the Minister for Magic, whose election is a matter only for the magical community themselves. All matters relating to the magical community in Britain are managed solely by the Minister for Magic, and he has sole jurisdiction over his Ministry. Emergency visits to the Muggle Prime Minister by the Minister for Magic are announced by a portrait of Ulick Gamp (first Minister for Magic) that hangs in the Muggle Prime Minister's study in number 10 Downing Street
I would presume that this jurisdiction extends over the entirety of Witches and Wizards, from all the conversations we witness between the Muggle PM and the MoM it would seem that the PM is not involved in anything.
We also get more details on Ulick Gamp:
Minister: Ulick Gamp
Term of Office: 1707 - 1718
Previously head of the Wizengamot, Gamp had the onerous job of policing a fractious and frightened community adjusting to the imposition of the International Statute of Secrecy. His greatest legacy was to found the Department of Magical Law Enforcement.
Notice that he founded the Department of Magical Law Enforcement.
I would however say that Muggle Laws (at least pre Muggle Magic separation) will have influenced Magical Laws.
As an example look at Sirius, he killed 10 muggles (as well as Pettigrew) and the muggles didn't interfere, he is punished solely inside the Magical Community.