My son has a terrible cold. Trying not to be too gross, but he has completely surpassed the ablility to use handkerchiefs. I don't have that many, and I don't want to get that many, because most of the year we don't need them.
You could use rags as a tissue substitute. I saved the fabric from an old pair of sweatpants and made a stack of handkerchiefs from them. Not the prettiest looking things but they get the job done (the terry cloth is absorbent and gentle on the nose). You can sterilize the used rags by boiling them in a pot of water for 10 minutes.
I have a 9 month old who just got over a cold. Instead of disposable tissues, I used the same thin cotton "paper towels" that I use to wipe his face after meals. I run them under warm water in the sink first. It might be my imagination but I think don't wipe as aggressively as I would if I were using disposable tissues. The water helps dissolve all of the dried on boogers (or food).
Here's where I bought mine: