Law enforcement will be handled by the National Guard. While this might sound authoritarian, the reason behind this is to have a standard of training that would be the same throughout the nation. The National Guard officer charged with law enforcement will have clearly police uniforms, and police style automobiles. The weapons used will also be policed issued. This means that a law enforcement officer is not going to carry an assault weapons unless they are in the SWAT division.
Law enforcement will need vehicles. If there is a national auto manufacturer, then they will get preferential treatment when bidding for new vehicles. A wide range of vehicles will likely be needed. A city car might be used for ticketing a car with a parking violation. A hatchback might roam the city streets to be a presence to the populace. The standard vehicles will be sedans and SUVs where there will be a sufficient size back seat for arresting people. A van might be used to take in multiple people into custody, or moving inmates from jail to prison, or their court hearings. Pickup trucks might be used in rural or remote locations making them capable of going off-road.
If there is an arrest to be made, then law enforcement will be responsible for doing the arrest. They will read the rights to the defendant, and place handcuffs on the wrists with the arms on behind the person being arrested. They will have a firearm as expected, but the firearm will have a biometric lock preventing anyone except the officer from using the firearm. This prevents use if the criminal acquires the gun. They will also carry a taser which will be preferred to be used if non-lethal force is to be used.
Police officers will have the same uniform throughout the nation. This will allow for simplicity and consistency. A 3 year old will learn that they can walk up to any police officer for help. They don’t have to be able to read, but they will know by the uniform.
For serious issues such as a hostage crisis, there will be SWAT (Special Weapons And Tactics) which will come in. Their job is to take down the assailants (lethally if necessary), and protect the innocent. Out of the law enforcement divisions, SWAT is the only one with assignments allowing for assault weapons. They will also have drones, and military style vehicles. This is not the norm however. For most interactions with police, it will feel more like a civil establishment rather than military trained officers.