尿にょう意いを催もよおす

Meanings

1

expression, godan verb (-su)

feel the urge to urinate

A formal or clinical expression for needing to urinate. In casual speech, people more often say おしっこがしたい or トイレに行きたい.

urine

意idea, mind, heart

催sponsor, hold (a meeting), give (a dinner)

便べん意いを催もよおす
Means 'to feel the urge to defecate'; the same construction with 便意 (need to defecate) instead of 尿意.
催もよおす
The verb 催す means 'to hold (an event)' or 'to feel (an urge)'. In this expression it takes the latter meaning.

尿意 (urinary urge) + を + 催す (to feel an urge). A straightforward compound using the formal noun 尿意.