Most cough and cattarh are usually due to viral infection which will normally run their course with or without any form of treatment.
For blocked nostrils. Ensure baby is well breastfed, kept warm, avoid frequent baths for newborns. Saline Nasal drops are over the counter drugs that can be dropped into the nostrils to soften the mucus and allow it to flow out freely.
You could also see the doctor for a follow up

Please continue breastfeeding your baby