Twee isn't exactly new. It started in the early '80s with C-86 bands, like the Pastels and the Wedding Present. The jangly strummy la la pop sound continued to grow with Sarah Records, which brought us the sweet, sweet sounds of 14 Iced Bears, Heavenly, and Talulah Gosh. Around the '90s, twee made its way to the colonies by way of K Records. We can thank K Records for the Beat Happening, Marine Research, the Softies, and the All Girl Summer Fun Band.
I don't think the term "twee" was coined until the mid-nineties-ish. I am reasonably sure that the late John Peel came up with that one. British DJs are always coming up with all sorts of terminology.