You know, lots of children wake up congested for not much reason at all -and it is doubtful that he is allergic to your cats as he would have had a reaction to them long ago (and allergies know no night and day. Its a 24 hour thing).
My daughter will wake up congested if her room is too cold or if there happens to be a breeze coming in through the cracks of the windows. Have you had your sons ears/sinuses looked at by his pediatrician? Congestion is often related, and even caused by, problems with the sinuses (they can be slow to develope properly in young children) and the ears. If you haven't had him seen by his ped, do that first just to rule out a sinus infection or fluid in the ears.
As far as treating the stuffy nose now - put a cool air humidifier in his room at night - he will find the white noise VERY soothing, and will clear up his congestion. If this doesn't help (but it usually does) you can try giving him the recommended dosage of infant's benadryll - if that does the trick, he is probably allergic to something in the house.