No one's entirely sure how long it will take the disposable nappies that have been used over the last few decades to breakdown in landfill. The estimate is that it will take each disposable nappy 250-500 years to decompose*, depending on the landfill conditions. That's long after our children, grandchildren and great grandchildren are gone.
To put it in perspective: if Queen Elizabeth I of England (born 1533) has worn disposable nappies, it's likely that archaeologists would still be digging them up!
So, do future archaeologists a favour, and use cloth nappies when you can!