What kind of Martial Art do US Military and US Police use?

They both use a combination of several martial and fighting arts really and there is a distinct difference between the two really. That which various academies teach law enforcement are different because of "allowable force" and the limitations that levies on departments and their officers. There are indeed things they are precluded and restricted from doing which is why you almost will never see for instance an officer striking to the head, knees, and elbows of a person but instead to fleshy, muscle covered areas with their belly clubs or side handle batons.

The military though does not have quite the same limitations and so some of what they teach is and can be more lethal in some ways. Some of it also revolves more around they or their aggressor having a long barrel weapon also while that of civilian departments almost never addresses this directly at all. The Army and the Marine Corps have both modified their combative courses a couple of times the past ten years or so. For a while they became so heavily oriented towards BJJ (especially the Army) that they soon found that individuals might instead rely too heavily on wrestling and grappling instead of maintaining control of their weapons and employing them to dispatch an individual while also not being able to lend fire support to others in their squad or combat team. So they scaled BJJ back in their program in their basic combative course for this reason after a few years of really embracing it.

Here's a question for you and I bet you can't guess which military service actually decided to bring in outside instructors and trainers from martial and fighting arts to develop a structured combative program to be taught to its members? The answer was the Air Force in 1960-1961 by General Curtis LeMay and he actually had the Air Force contract with several notable Japanese instructors for that purpose and helping to institute a structured training program. It later fell by the wayside with his retirement in 1965 but by then the Army and Marine Corps had picked up on the idea also and with Vietnam looming they in turn started their own programs.

The US military has their own unarmed combat training call combatives which are a combination of many martial art where the most relevant techniques are borrowed and fused along with the combat experiences from over 200 years of conflict. The focus is on practical applications and not for sports or competition. The training is also distilled in such a way that a recruit can be trained quickly in the shortest amount of time because the primary focus of the modern day military is not on unarmed combat, but high tech weaponry.

There is no US Police. Each state and city in the United States has their own Law Enforcement agency which have their own training programs. There are Federal Law Enforcement agencies who have their programs, too, including sending their agents to be trained by the military using the "combatives".

Marines train in MCMAP

https://en.wikipedia.org/wiki/Marine_Corps_Martial...

Army trains in Army Combatives

https://en.wikipedia.org/wiki/Combatives#External_...

Air Force trains in Air Force Combatives

https://en.wikipedia.org/wiki/Combatives#Air_Force...

NAVY SEALS train in Executive Progressive Fighting Systems (Jeet Kune Do Concepts & Filipino Arts)

http://fighting.net/

There is no governing body for cops some learn Krav Maga some learn Jeet Kune Do some learn other arts

There is not one uniform style of instruction. I do not have current data, but in the past, the US Army did send promising individuals to train in TKD with the Korean Army.

Kali, catch wrestling, and FMA in general.