Hmm, I tried using non-changing bags (a selection of ineffectual laptop or beach bags before I gave up and bought a proper one that is made by the same people are yummy mummy but without the sickening slogan), and when they are small, if you are breast feeding and if you only go out for short periods of time, preferably in the car, then it's perfectly do-able to do without.
I would suggest you have a bag you designate as the baby bag (even if it isn't an offical changing bag) that you keep stocked with nappies, wipes, nappy sacks, mussies, bibs (if you have a dribbly one), a full change of outfit for the baby. If you end up FFing make sure it has space for bottles and formula (and if you use the ready made cartons, pack sissors, if you don't, make sure you can store the bottles up right so the water doesn't all fall out) - makes leaving the house a lot quicker!
And when they are weaned, make sure it's got space for baby food/snacks. Ideally, make sure it's a bag that has a strap big enough to fit over the handlebars of your pram to avoid having to lug it about/fill up your basket under the pram with it (so you have nowhere to put your shopping).
Most woman don't already own a bag that fits the bill, so end up buying a changing bag.
Mind you, the free one you get from the Boots parenting club fits the bill too. (and it's plain black and slogan free)