Hi, welcome to the forum. It is quite important to rule out the cause of the cough, at the same time exacerbating factors, such as exposure to environmental tobacco smoke, dust etc. should be pointed out and further exposure should be avoided.
It is important to know whether he has aspirated any foreign body. The type of cough in his case suggests cough dominant asthma. This may be seen in association with the factors like cold air, sleep, or allergens.
The other causes needed to be ruled out are airway malacia, laryngotracheobronchitis and infection with Chlamydia trachomatis. If there is presence of sputum with cough then we can suspect the infection.
He needs to undergo certain tests like allergen test, spirometry, and serological test for Chlamydia and bronchoscopy to rule out foreign body.
Treatment includes cough suppressants, a trial therapy with bronchodilators, avoiding triggering factors, and available home remedies like use of honey and echinacea.
I suggest you to consult pediatrician specialized in respiratory medicine for evaluation and prescription and above drugs. Take care and regards.
Since you have a nebulizer, I am asumming someone in the family has asthma. If he has asthma he is probably in a good flare and needs some controller meds and possibly some prednisone to bring it under control. What did you use in the nebulizer for him? Pulmacort or albuterol? Pulmacort could take quite a while on its own and albuterol without Pulmacort would be very temporary.
Also, if he is experiencing allergies you need to be treating them with antihistamines. Also, he is old enough to learn how to do saline sinus rinses. You can get a NeilMed bottle at any pharmacy. That is very helpful in clearing the gunk out of the nose before it causes other problems.