Most units are actually just numbers, so really you would just need to change the name of those numbers and the names of nations.
Its really simply. But most people wouldn't enjoy modern warfare, as most weapons of wars (naval weapons for example) have become more advanced, but shrunk in terms of different types of said weapon.
For example, while Aircraft Carriers have become dominate in the fleet, Battleships and battlecruisers (save the Russian Kirov Class) have been dropped. Your navy would consist of three surface ships, and submarines. Your tanks would be either Main Battle Tanks, or light IFV.
All infantry today is either mechanized infantry, marine or paratrooper. Thats three types of infantry only.
Fighters can now carry out bombing missions- thus being called "Multi-role fighters" only the US/Russia/China keep large stocks of cold war era bombers. So our air force wouldn't be very large.
You see the issue? We'd have few units to play with.