All you need for a nappy bag is some form of a tote bag, well as far as I know anyway. I have never actually looked inside a specifically designed nappy bag so I'm not sure what makes them different to a tote bag. You do need a bag so I say take the opportunity to buy something you love. Here are some options in extremely varying price ranges. I have a different version of the Company of Strangers bag below in dark blue, it is made from recycled leather jackets.
Clockwise from top left: Company of Strangers, Alexander Wang, MM6 by Martin Margiela & Something Else by Natalie Wood.