Is that a plausible result of Charlotte living? that'd be cool...
Well, Phil Masters (the author of the piece) suggests that - as heirs to the throne - a surviving Charlotte & her husband are seen "as a refreshing and popular alternative to the aging and increasingly reactionary royal generation preceding them." They're courted by the Whigs (with Charlotte a known Whig sympathiser), and - in cementing an informal alliance of sorts - the death of George III triggers an election in which the party do a lot better than in OTL, gaining enough seats to form a government.
Slowly, Charlotte and the Whigs manage to talk round George IV to their way of thinking, and a bunch of the later liberal and reformist policies of OTL (electoral reform, a shift to free trade, enfranchisement of religious minorities) come a couple of years ealier.
Following George's death, Charlotte and Leopold ride the wave of popular liberalism and the technological advances offered by the Industrial Revolution, with the increasingly pro-Whig "bright young things" of the British monarchy (and their hangers-on) far happier to engage socially with the "new money" of industry. Aristocracy and plutocracy intermingle, with the new ruling class demonstrating "a truly startling [slightly geekish?] enthusiasm for science and especially technology."
In OTL, Leopold leapt at the chance of spearheading a Belgian railway relatively early on ... and, with Charlotte's survival, he's in a relatively influential position to cement a "royal approval" for backing research, particularly into shiny "steam carriages", bridges and buildings, war mechanics, and weird and wonderful experiments in flight. A lot more in the way of scientific competitions with cash prizes, amusing bets and gambling, and (with royal assent) professional societies for scientists and engineers.
With this technology at its disposal, C19th the British Empire is a lot bigger and more influential that in OTL, with other powers attempting to mimic its techno-liberal recipe for success. Oh, and with American "Jacksonians" increasingly suspicious of a close relationship between the British Whigs and President John Q. Adams and his supporters, an increasingly weathy, liberal and industrial New England secedes from the rest of the US (with Canadian support) in the 1830s/40s, to form some kind of technocratic inustrial state.