I've never had a nappy bag. With DD1 I had a backpack which worked well as I walked a lot and often shoved shopping etc in there as well.

With DD2, I had a $15 satchel from a bag shop, it's got three main compartments plus some smaller ones and it's just the right size to fit the necessities, including snacks & drinks & hats, etc. The strap is extendable and fits right over both of our pram handles.