Politer ways to say "to fart" include "to pass gas", "to cut one", "to pass wind", etc.
The boy just had an accident in his diaper, dropped a load, took a dump, made a stinky, etc.
The girl went wee-wee in her diaper.
To piss, to sh*t and to fart are rather strong words. I suggest you not use them in polite company.
"Toot" can be used in polite circles.
"To have a pee" implies that the person stopped what he was doing, went to the bathroom (or to a place behind some bushes), and urinated. It implies much more than "to urinate".