Taylor's communication with all of the various space agencies is near-instantaneous. There is not the delay in transmission/reception and re-transmission that there would be with a spacecraft in low orbit.
After Taylor sacrifices his cabin heaters to restore power to the rest of his equipment, the capsule's interior temperature drops within eight minutes to near-freezing temperatures. Actually, manned spacecraft require internal cooling, not heating, even when passing through Earth's shadow. More heat is generated by occupants and equipment than is dissipated to space in the form of electromagnetic radiation. Even during Apollo 13, in which nearly all equipment was turned off to conserve the crippled spacecraft's power, similar temperature loss occurred over the course of days, not minutes.
At the end of the film, the Russian soldiers are wearing navy jeans or slacks, contradicting the rest of the traditional olive uniform.
The Russian military vehicles at the end of the film have no serial numbers nor national insignia.
The Johnson Space Center (aka Houston Control) was opened for business in late 1963, nearly 5 years after the setting of the film. The control centers at Cape Canaveral, Florida or Langley, Virginia would have been the US contact points for any craft that was orbiting the Earth prior to 1963, as "Houston" did not exist.
Taylor is contacted by NASA when his craft overflies the United States. While NASA was established by President Dwight Eisenhower in 1958 ( the year before the film's setting) the agency didn't take over the US spaceflight and launch role until early October of 1958.
It's unlikely that Taylor would have known what NASA was and he would have likely had to have its role explained to him. He would have likely expected contact from National Advisory Committee for Aeronautics, NASA's predecessor.
During a close-up scene of Guy near the climax of the movie, it can be seen that he is wearing soft-lens contacts, which did not exist in 1959.