Does a severe cold and cough 1 month after hair transplant affect the outcome?

It is best to ask your surgeon about post operative care as each doctor has their unique regimen.  If you have a severe cold, see a doctor for treatment.  It should not affect your surgery.

At this point (or even earlier in the process) a cold or cough should not have any adverse effect on the grafts or the ultimate result.

At one month, everything should be secure and coughing or the flu should not impact the growth process