No problem with the southern boundary, which can be nothing but the river. In the west we come to more water, with Chelsea turning into Fulham and the World's End into Sand's End at Chelsea Creek, also known as Counter's Creek, a mainly subterranean stream which rises in Kensal Green and flows through Olympia and West Brompton Stations, in between Brompton Cemetery and Stamford Bridge (the stadium), under Stamford Bridge (the bridge) and Stanley Bridge and into the Thames between Lots Road Power Station and Chelsea Harbour. In 1859-63 much of it was culverted and the West London Line was built along its course. The Creek and the railway part company near the southern corner of Lots Road, running north and south of Chelsea Harbour respectively, the former into the Thames and the latter across Battersea Railway Bridge.
The northern boundary of western Chelsea runs straight down the middle of the Fulham Road, the north side being in Kensington; in fact most people would not consider Fulham Road to be World's End, which is felt to be situated mainly between the King's Road and the river. On the other hand, no one would deny that Slaidburn Street, running north off the King's Road, belonged to the World's End. Basically, the further away from King's Road and the closer to Fulham Road you get, the less likely you are to call it World's End.
A similar concept holds true for the eastward extension. Whereas most people would include the S-bend at Moravian Corner and Milman's Street, others might go as far as Beaufort Street and Battersea Bridge.
I'd always thought the World's End corresponded to the postal district of SW10, until I looked at the street signs and found out that SW10 extended to the north as far as the Old Brompton Road. The name corresponding to SW10 is West Brompton, so the World's End can be said to be the southern half of West Brompton.
The definite article is optional in the name of (the) World's End, as it is in (the) King's Road and (the) Fulham Road. Through a series of Google searches, having cleverly eliminated the millions of references to Pirates of the Caribbean: At World's End along with William Morris's The Well at the World's End, I discovered that the World's End and the King's Road are more often found WITH the article, Fulham Road WITHOUT. On the other hand, the expression "at/in World's End, Chelsea" is much more widespread than "at/in the World's End, Chelsea", which hardly appears at all.
Until 2002 World's End south of the King's Road corresponded to the South Stanley ward, named after Stanley House and Stanley Bridge on the border with Fulham, electing two (Labour) councillors to the Royal Borough of Kensington & Chelsea (before 1965 to the Metropolitan Borough of Chelsea). North Stanley was north of the King's Road. Since 2002 South Stanley has been incorporated into the new Cremorne ward.