I wonder if the Nike Ajax used any gyros at all, if the radar guidance
uplink was continuous from launch.

Radar guided from launch.

I imagine The Military Masterminds would not have expected the 'Bears' to be
accompanied by radar suppressing 'Thuds': "Fishboobs".

The missle was headed away from the launch site and toward the target.
The receiving "horn" was directional enough to maintain a good S/N
ratio in the face of target aircraft jamming efforts.

Also, what kind of propellant/oxidant was used?

It was a two-stage missle with a solid fuel booster and a liquid fuel
missle motor. One of the liquid components was "red fuming nitric
acid" but I forgot the other component. It's been well over 40 years
ago, so cut me some slack!

Also, was the uplink data stream encrypted?

No.

What kind of warhead did the nike Ajax have?

The Ajax had three warheads. A 12 lb. nose warhead, a 179 lb. center
warhead, and a 122 lb. tail warhead. The warheads were TNT and
covered with layers of steel blocks about 1/2-inch square. The idea
was to produce a burst pattern of sufficient size to cover the target
aircraft.

Best Regards,

Ed Engelken
Canyon Lake, TX