In the case of "New York's Finest," Barry [Popik] has traced the term back to
the 1870s, where it apparently first emerged in the form "the finest
police force in the world," a phrase associated with George Matsell
(police chief at the time), and possibly modeled on Civil War Major
General Joseph Hooker's estimation of his troops as "the finest army
on the planet." The general idea of New York's police being "the finest" had been asserted in print for several years by that point, so it
wasn't surprising that eventually the tribute was codified in the
phrase "New York's Finest."
I've never heard the police in London referred to as "London's finest", and a search for that phrase results in must-see landmarks, expensive hotels, hairdressers, dance troupes and man-and-van services. But adding the word police shows the Metropolitan Police do (or have) used the phrase:
A career with London's finest
I don't think it's an official term in London, unlike New York and some other American cities.