It should be first noted that nuclear weapons were not manufactured in New Mexico. Actual manufacture of stockpile weapons was done at the Pantex plant in Texas, from the early 1950s onwards. However fissile material was used for experiments and the like in New Mexico.

I don't exactly know about the time period you are asking about. In the late 1960s/early 1970s movement of fissile material was done by truck or by special rail cars. The trucks moved in small convoys and had radios in them. The truck drivers would also stop to use pay phones to check in on a regular basis. The rail cars were armored and had guards in them. They relied on being discreet more than heavy armor. The fear of hijacking of these trucks, etc., did not really come up until the 1970s, when domestic and international terrorism began to increase. By the mid 1970s these methods were seen as insecure. On this latter point, see J. Samuel Walker, "Regulating against Nuclear Terrorism: The Domestic Safeguards Issue, 1970-1979," Technology and Culture 42, No. 1 (2001), pp. 107-132.

Small amounts could be sent by airplane. In 1972 these were set at well under a kilogram of fissile material.