I think brass instruments actually made a comeback via the freak folk scene in the Noughties. And it wasn't unusual for indie bands to feature the odd trumpet, saxophone or even trombone players (e.g. Broken Social Scene) - at least, more so than in the Eighties or Nineties. And there also has been a recent tendency to use the saxophone in dance tracks again. I'd say that this may be partially a result of a younger generation of musicians who used to play in marching bands later using their skills for doing their own thang, e.g. ending up in rock/indie/punk/dubstep acts. So you'd just need to amplify this tendency. I had this fantasy idea for a European answer to freak folk: young kids from Austria/Switzerland who grew up listening to indie, dance or punk start to incorporate the traditional instruments (including, for instance, the alphorn) of their upbringing.
Otherwise, you could just make ska or rocksteady a big thing in the Sixties. If ska went as big as soul, beat or psychedelia, then you could develop a scenario where a trumpet player is a natural requirement for bands like bassists or guitarists. Or, instead of only using backing bands, soul acts with a continuous horns section evolve, thereby making brass solo players as important as singers or guitarists.