If wearing nappies gained widespread acceptance I think it would make life a bit easier and nicer for us; I can't see how it would take any of the fun out. As things are today, I wear nappies instead of using the toilet because it feels better, they are comfy and convenient and it just makes me happy. It would be much the same if nappies were more popular, except that the convenience factors would improve. Bathroom stalls would be made bigger and unisex bathrooms would probably be more common so that couples could change each other. There would be more high-street stores with more variety, including fashion-oriented products and accessories. You could discuss absorbency and fit over coffee with friends and try each other's favourite brands.
It's difficult to predict the social mores that might develop once the awkwardness is stripped away - some might be quite radical. If you compare toilet etiquette in different cultures, there's quite a variety of different expectations of modesty. In China, for example, there have traditionally been toilets with zero privacy, where men and women alike just squat over a trough to wee and poo in front of friends and strangers. Perhaps the same openness would develop about nappy changes, triggered by the greater acceptance of one's own and each other's needs. You would just lie on a changing table in an open room and change / be changed. Perhaps it would be OK / polite to offer to change your friends if you go together. Even in public, there might be a thing for showing off your nappy e.g. going out with just a nappy and top on, particularly if you wear nappies with fashionable designs or just to keep cool in hot weather, and not be embarrassed about the wetness showing. If you felt like a change, you could just look at your friends' wetness indicators to see if they need changing too and invite them along. It could be sociable and fun, provided the necessary hygiene factors were also part of the routine.
If there were a switch I could flick to make this all a reality, I would go for it, but even small steps would be enjoyable. Maybe in 10 years it will be OK to tell friends that you're going to wear a nappy to an event, or ask if they are wearing, without getting any more open than that. Surely there's nothing to lose?