A month of cough with wheezing and breathlessness on exertion often points to asthma or an airway that's become reactive after a recent infection. This is common and treatable. The bluish tinge in your nails when you're cold is usually a separate, benign reaction called Raynaud's, where small blood vessels narrow in the cold. It's unlikely connected to your breathing unless it happens even when you're warm.
Next Steps
A pulmonologist or physician can confirm asthma with a simple breathing test called spirometry, and can recommend an inhaler suited to you. Keep track of what triggers your wheezing, such as dust, exercise or weather change, and mention it at your visit.
Health Tips
See a doctor soon if your breathlessness worsens at rest, your lips or nails turn blue without the cold, or you feel too breathless to speak in full sentences.