Like bonkers and a piece of cake, this word for “fashionable” or “voguish” doesn’t sound like a Britishism, but it is.
It seems to have popped up in London in the early 1960s; the OED’s first cite is a supercilious quote from a 1962 Punch: “I saw the headline ‘The Trendiest Twin Set’.”
The first use in the New York Times came in 1968, via the paper’s British-born art critic John Russell. It took hold quickly, because the following year, reporter Steven V. Roberts referred to Roman Polanski as being “been near the center of a loose group of film makers who were described with all the current cliches: rood, hip, swinging, trendy.”
As this Google Ngram Viewer chart shows, within about fifteen years, U.S use had surpassed British use, never to look back:
notoneoffbritishisms.com 
I remember hearing it first on the BBC radio show (I won’t say “programme”) “Round the Horne,” a sort of descendant of the Goons and home of “I’m Julian and this is my friend Sandy. OOOH, it’s Mister ‘Orne. How nice to varda your old eek again!” and the dearly departed Kenneth Williams. “Trendy” was usually coupled with “with it.”
A friend refers to a fashionable boomer/yuppie Nashville suburb as “tragically trendy.”
So an outside possibility it’s from Polari (or maybe just general theatrical slang?) and was brought to wider attention courtesy of Cryer, Took and the Home Service? Might explain why it doesn’t appear widely much before 1964/5.
Maybe, but I seem to remember Kenneth Horne as the one who used the word. Something about “for the trendy, with it … ” people who would like what was going to be presented in the show. Usually at the beginning of the programme.
My understanding is that ITMA ran from 1939 to 1949, so that seems a bit early to be a precursor.
Well, more a descendant of ITMA, perhaps.
My wife tells me “trendy” is no longer trendy in Britain. You are now supposed to say “in trend” if you want to stay in trend.