It was Cuba, not Russia. Russia merely sent advisors and Wolverine hunters, not an invading army.
"The film depicts a fictional World War III centering on a military invasion of the United States by an alliance of Soviet, Warsaw Pact, and Communist Latin American states."
Russia is the main invasion force, with Cuban and central American auxiliary units. It’s explained in the film by Col Tanner, and he also detailed the nuclear exchange by the Soviets.
Remember Powers Booth describing the course of the war. The Russians crossed rhe Bering Strait and drove all the way to Montana. Plus th first two guys the kids kill are young Russian soldiers