The colloquial phrase down the Swanee means completely lost or wasted—synonymous phrases: down the pan – down the toilet – down the tube(s) – down the plughole – down the drain – down the gurgler.
What does up the Swanee mean?
Up the swanny / up the Swanee is indeed a British-English term meaning "in serious trouble", in a probably permanent way, e.g. pregnancy, the "big end" failing on one's car, the British economy in 2008, being sent on a suicidal military mission by one's uncomprehending superior officer, etc.
Where did the saying I'll Swanee come from?
An 11-year-old in Tallahassee, Florida, wonders about a phrase her late grandfather used. Instead of swearing, he'd exclaim “I swanee!” or “I'll swanny!” This mild oath, and its shorter version, “I'll swan,” derives from an English dialectal phrase, “I shall warrant.” This is part of a complete episode.