In terms of music culture, Wagner was responsible for how the concert and concert hall were developed/viewed, and the whole 'aura of awe' around performances and performers. This would probably mean that Western art music would feel more accessible to people.
Brahms would definitely be considered 'the' German composer of the era. While as a musician I love performing Brahms more than Wagner, I don't think it's possible for Brahms to have the same kind of impact that Wagner had -he wasn't really revolutionary, he didn't write about the arts. But we might see absolute music rather than programme music being more popular, at least in Germany. The entire idea of gesamtkunstwerk is Wagner's -the combination of all the arts into one. Berlioz might take the place of Wagner, as the champion of programme music.
It would lead to radically different compositional styles for many composers, notably Mahler (and Richard Strauss). You could also say he was the real precursor to the post-tonal composers. Then there were the composers who wrote in reaction against him, like Debussy, and I guess the Impressionist school.
The use of chromaticism, and thus post-tonal music, as mentioned above, would be very different. I mean, Chopin used a lot of chromaticism, but he didn't do orchestral stuff. In general, you have to have 'big' pieces, i.e. large ensemble pieces, to make an impact in the Romantic era (then again, Schubert didn't really do orchestral stuff, and he was very influential, but he was right near the beginning of the Romantic era, and the atmosphere was different, but feel free to argue with me, this isn't my area of expertise). So all those many, many, composers who heard Wagner and started writing chromatic stuff aren't going to.
I do think that music to some extent reflects trends in society, and chromaticism represented a dissatisfaction, and the general idea of revolution beneath the surface (I think the term used was malaise du siecle), so it would have popped up at some point in the 19th century. And it would have been in Germany/Austria. Not necessarily because there was more dissatisfaction there, but because in Western classical music there are basically three schools: German, French and Italian. French music, I think, had its own way of dealing with this malaise -my impression (ha ha) is that they went more after form than tonality. I've also never really gotten the impression from Italian Romantic music that they were that interested in chromaticism or musical progress, so to speak -could be because they felt more entrenched in their musical history? Whereas German music from the Baroque period on was never considered quite as prestigious as Italian, or even French. So they've got something to prove. They've got problems with society. They need something to call their own. And they have the music education system and environment in which you can get performed. (I'm ignoring Russia because they didn't really get started until after Wagner). I don't think it's entirely a coincidence that tonality got tied up in nationalism, or that most of the composers who pushed tonality from this era were German. I can't for the life of me figure out who else could do it, since most of the likely candidates (Bruckner? Strauss?...Mahler?) were ridiculously influenced by Wagner. I'd be really interested to see how post-tonal music developed in that world, though.
As said, Verdi would have become the reference of late XIX.
It would have meant a more prevalent place of opera and operette in classical music of this era, more lyric than epic.
Furthermore, I think Wagner did more to preserve musical choices of XIX : don't get me wrong, he uses them differently. But without Wagner, maybe we would have got an earlier modernism in classical music while having less "national" music (or more close to Verdi conception than Wagnerian, less "folkish")
AFAIK, Wagner didn't use folk music? The 'Big Three' didn't really tend to go the folk song route, because they had all that art music tradition. It was mainly the ones that didn't, like Poland, Russia, etc., that drew from their folk music. But I do agree that without Wagner, music nationalism (read: German music nationalism) probably wouldn't have gotten that 'aggressive'. Some of it was also due to Nietzsche, but he was seriously influenced by Wagner, so if we take that away...hmm. That could lead to some nice political consequences.