Hi M.
I experienced the same thing when my son was at that age.
They tested for RSV and fortunatly he didn't have that. As they are too young at this age to be diagnosed with Asthma, our pediatrician said it sounded like RAD (Reactive Airway Disease).
He was treated with Albuterol and Prolicort (use of a nebulaizer) for about 2 months, along with Prednosolone
After the 2 months, the wheezing, excessive coughing and all of the other effects disappeared.
Once in a while during and after a cold, we have to bring out the nebulizer to control the RAD. We normally would do it twice a day until it went away, which was about 2 days.
Let us know what you find out.