I spent a considerable amount of time and effort looking for some non-anecdotal information on this question and while I found nothing that seemed believable, my assessment is that the available W7 warheads were probably stored at the special weapons magazine in Fort Lewis and not dispersed to the firing batteries unless necessary. Since the war portrayed in Deter, Detect, Defend starts without warning, only the conventional warheads are used. There are a number of reasons for this:
1. In 1962 the armed services were in a continual battle over custody of nuclear weapons with the Department of Energy (DoE) responsible for the manufacture and storage of all nuclear weapons and components that had not been allocated to direct military control. Most Nike Hercules batteries were National Guard with the high-readiness installations in Alaska and south Florida as exceptions. For the most part it appears that the custodial requirements for storage and maintenance of nuclear material exceeded what might be practical for most NG units with regards to infrastructure, training and personnel qualifications.
2. In December 1962, President Kennedy vetoed a DoD plan to increase the dispersal of warheads from DoE to DoD custody. Since clawing the weapons back from the services seems unlikely in this era, presumably they had not yet been delivered to the military. In any event, SAC and the Navy probably had priority over the Army in the CONUS.
3. The presence in nearby Fort Lewis of appropriate special weapons storage facilities and technical expertise makes dispersal to the Puget Sound Hercules firing batteries expensive and arguably unnecessary since in a crisis, warheads and personal could be quickly deployed under Regular Army control.
4. Information indicates that only 2500 W7 warheads were manufactured for some 25,000 MIM-14B missiles. This means that even under ideal circumstances nine out of ten missiles would be conventionally armed.
5. The 1960 BOMARC accident in New Jersey pointed to the hazard of nuclear warheads mated to live missiles near urban centres.
My assessment for designing the scenario was that the Puget Sound Hercules firing batteries were probably equipped with conventional warheads at this time. Spent considerable time in The Nuclear Vault (nsarchive.gwu.edu) were there is a large document archive but much of the relevant material is still redacted to uselessness.