What's the word for the condition where you need to defecate?

There is no common word that describes this condition. American English speakers will politely say:

I need to go to the bathroom.

British English speakers may say something like:

I need to go to the loo.

I need to visit the WC.

None of these are explicit about what kind of waste is being eliminated, but in polite company you don't usually need to be exact about your toilet activities.

There are a wide variety of less polite alternatives, which I started to try to list, but gave up.

My grandmother would never be more explicit than to say she "needed to powder her nose". The fact that she didn't wear makeup was quite beside the fact. On the other end of the spectrum, others might have a "need to take a dump".

Fecal urgency is the term for really needing to go.