سلام دوست عزیز.
در جواب سوال شما اول باید بگم که کمتر پیش میاد در حالت شماره یک (موقعیت رسمی و مودبانه) از کسی چنین سوالی پرسیده شود، مگر اینکه مورد پزشکی باشه. اما موارد دو و سه امکانپذیر است. با توجه به این توضیح، اینها جوابهای من هستند:
#1
-A: Sorry I need to go to the restroom (AmE)/ toilet (BrE).
-A: Sorry I need to go to the men's room (AmE)/ gents (BrE).
-A: Sorry I need to go to the ladies' room (AmE)/ ladies (BrE).
-A: Sorry I need to use the facilities.
-B: Will it take long or ...? (this way you avoid using any relater words)
-B: Is it number one or number two? (number1=urinate, number2=defecate)
-B: Are you gonna urinate or move your bowels? (said by a doctor)
#2&3
-A: I have to go to the bathroom.
-A: I have to answer the nature's call.
-A: I have to see a man about a horse.
-A: I have to see John.
-B: Is it number one or number two? *
-B: Is it pee or poop (AmE)/ poo (BrE)? *
-B: Are you gonna take a leak or take a shit? **
-B: Piss or shit? **
* These expressions were once used by adults when talking to children, but later on they found their way to adults' speech too.
** These expressions are not at all polite and just used by adults in friendly and informal gatherings.
P.S. In the second category, only the first sentence (I have to go to the bathroom) is said by children, and the rest are humorous ways of saying that, used by adults.