Healing rate of cut while suffering from a cold
First of all: No, there is no connection between these two incidents, correlation does not imply a causation. Cold and flu viruses affect the respiratory system and the infection route goes via it. A cut is a local wound, which does not affect the respiratory tract.

Then: Recovering from an infection is a task for the immune system (its both, the innate and the adaptive immune system, the last involves the antibodies), which is not doing much for wound healing. The wound attracts neutrophils and macrophages which are attracted by factors released by the wound and they react to bacterial infections (when they occur). See this Wikipedia article. 

The ressources of a normally healthy person are quite big. This is demonstrated by the fact that many secondary infections only occur in a opportunistic fashion (for example viruses like Herpes or certain fungi) when the body is weakened by other infections. So there is a limit, but this will certainly not be reached with a severe cold.