Is a common cold or the flu likely to reduce affect immune system sufficiently to allow the recurrence of a head/neck cancer?

No. The immune system is more likely to be affected by chemotherapy than by a cold or the flu.
No. Having a cold does not reduce the immune system. In fact, the whole problem with the common cold is that it is a disease in which the immune system over-enthusiastically attacks a mild virus, causing all of the miserable symptoms we know and love. If anything having a cold is a sign of a robust immune system and shouldn't have any implications with head or neck cancer.

Related Questions

Is It possible to be immune to the flu? Never had a flu shot in my life and I can't remember the last time I was sick other than a common cold..

Just lucky so far. There is no natural immunity to influenza. At least half of all people your age (20) have never had overt influenza, so this is nothing unusual. You have just been lucky so far; either you never caught it, or you had mild symptoms. Flu often is no worse than a common cold. You definitely still should always have recommended influenza immunization every year. Read more...