Ours just go in the change bag too

I've got a great bag from Boots that I use with the buggy. It's basically just got a divider down the middle, so I put clean nappies to the front and my change mat and dirty nappies go at the back. I do have some zipable plastic wallet things to put dirty nappies in, but 8 times out of 10 I forget one and end up wrapping used nappies up in my change mat
Most of the time though we use a back pack change bag by BabyLo, £7-something from TKMaxx

as DD is usually in an MT when we go out. This is just one big hole for me to chuck everything in

and the bottom zips round so thats where the used nappies go