The story
Unclear. As an idiom, identified only in Australian and New Zealand English. * Originally a sailing term for when the ship is taking in water and the ship is literally being bailed out to stay afloat. In the days of yore, the crew had to work 24 hours a day pumping out the ship until the ship reaches safe harbour for repairs. * Also possibly as a punishment. * Also suggested has been an origin in the construction industry: Workers can be quite literally under the pump if a concrete pour is occurring and the steel reinforcing and form-work is still being finished. The concrete pump would be reaching out above the workers, who would be completing their duties at a frantic pace. Once the concrete has been delivered, the ‘pour′ must occur as soon as possible, or it may begin to set. Hence, being under the pump describes the situation of working at feverish pace to meet a specific deadline.
Etymology from Wiktionary, CC BY-SA 4.0.