We have installed a Glacier Bay shower mixer/neck with a different shower head attached. In testing the water temperature to set it at a safe level, we found we are getting no cold water at all. Is there something we have missed in the install?

Do you have cold water up to the valve? If so it sounds like you dislodged some minerals built up in the cold water supply and it's now blocking the inlet ports of the cartridge,
Turn off the water and open up the valve. Check and clean the inlet ports on the cartridge.
Now turn the water back on for a moment to flush out the supplies.
Reassemble and test. Better now? Good luck, Tom