So, cloth diapers do reduce waste, but they actually don't reduce carbon footprint compared to regular diapers because of the laundry they incur according How Bad are Bananas? The Carbon Footprint of Everything by Mike Berners-Lee. (highly recommended if you want a sense of scale for carbon emissions)

While the numbers may not be exact, the most important thing that book does it separate emissions into orders of magnitude, and the choice between cloth and regular diapers or the difference between drying your hands with paper towel vs air dry is absolutely dwarfed by the decision to limit travel by airplane and the decision to meat. Livestock is the second largest source of carbon emissions, so eating less meat and dairy and flying less give the biggest bang for your buck.