There seems to be some connection between the Ministry of Magic and the Muggle Government of England.
There is some minimal interaction, but the Ministry of Magic functions as a de facto
independent government that isn't answerable to the muggle British government.
However it is not really fleshed out in the movies very much and I have to ask if there is any clarrification in the books?
Apparently, Ministers for Magic are supposed to meet with the Prime Minister and keep him informed about magical affairs that could affect the muggle populace, but canonically the extent of this informing was for Fudge first and Scrimgeour next to walk out of a fireplace, explain in the most obnoxious/patronizing form possible that yes, magic is real and walk out after commenting that there is a dangerous terrorist warlock running around.
For instance, the Ministry is overthrown by Voldemort so you have a coup in place. If there is a close relationship between the MoM and the Muggle English Government wouldn't the British Government be on high alert, perhaps even try to restore the old MoM with some well placed military pressure?
Prior to Voldemort taking over, the MoM was protecting the muggle goverment from Voldemort with infiltrated Aurors (including one acting as the Prime Minister's secretary), since some Death Eater got the bright idea to try to take over the muggles but botched the Imperius casting in the one minister they managed to get to. After the collapse of the legitimate wizarding goverment, it is a good bet that all relevant authorities were mind controlled into submission.
If not war, then one might be controlling the other for example in the opening scene of the movie Hermione erases all evidence of her existence from her parent's minds with the wave of a wand.
That's only in the movies.
In the books, she completely rewrites their personalities and sends them to Australia to keep them out of Voldemort's hands, happily unaware of the fact that they ever had had a daughter.
So with that in mind perhaps the wizarding world controls the muggle world subtly or not so subtly and that is why there is no conflict. If so what a sad little world to live in.
Pretty much. Canon: Fudge arranged for the muggle president of a foreign country to be mind-fucked so he could have his fifteen minutes with the British Prime Minister.
Also we naturally only see an English perspective, is there an American Dept of Magic that acts in the shadows in the US?
At least, Bulgaria and New Zealand have Ministries of Magic, as well. The government of other international parts of the wizarding world is unknown.
DeathEaters commit terrorist acts throughout the last few movies such as knocking down Milenium bridge or firing off death bolts on a highway. Sounds like the potential target for drone attacks in the making.
Thing is that while the "offensive firepower" of Potterverse wizards kind of sucks, they have excellent concealment magic, personal teleportation that cannot be tracked as the equivalent of a driving license and easy peachy magic mindfuckery. Regardless of who wins (and lets not open that can of worms, please), total or open war between mundanes and wizards would result in absolutely horrific damage all over the place.