Babies do require a nappy change more frequently when they are few weeks old, I am wondering is it a problem to change the diapers onboard? Do I have to worry about health issues when using lavatories? are lavatories equipped to handle this?
Changing on diapers on a plane is not a problem. Doing with no consideration for other passengers is a problem.
I can't speak for the quality of every airlines lavatories, but the feces from the baby probably present a greater health risk to others then whatever germs may be in the lavatory present to the baby. I'd recommend standard hygienic protocol (lots of wipes) and, if you're worried about the baby getting something in the lavatory, even wiping things down beforehand, .
Most airplane lavatories are equipped to handle this.
See also: Number 5 on this CNN list.