Not just browsers, but all image editors (natively) like Photoshop.
But at its core, WebP hasn't been shown to be more than marginally better than JPEG, so why bother modifying your tool chain and work flow? You generally need something to be an order of magnitude better for people to bother.
WebP support transparency with lossy compression which is a huge advantage over JPEG.
When I was studying image compression back at university we wrote a whole bunch of lossy/lossless codecs. The issue is that most of the cool algorithms are protected by patents (such as Arithmetic coding). JPEG is quite old now and you can achieve much better results with wavelet coding as opposed to the traditional DCT approach. Plus it doesn't suffer the horrible blockiness effect that quantized DCT based stuff does.
I was under the impression that Wavelets were something of a dead-end. As far as I picked it up the wavelets need to be tuned for your output. People have tuned them for great PSNR output but not for SSIM or other perceptual outputs so you get output that is blurry sometimes even compared with JPEG. I'm not sure if this is a fundamental problem with the technique, or just something that's currently impracticable compared with less elegant but working alternatives.
That is a big problem with the wavelet based approaches. They tend to be tuned for impressive PSNR numbers which doesn't always correspond well to perceptual image quality. I haven't seen that article in a while and he raised many valid points.
I think wavelets do have a future but it will just take more time and research. I'm not too sure what the state of it is at the moment as it has been a while since I've looked into it. It is my opinion that in time wavelet based compression will become more important though.
Edit: I forgot to mention that one really cool thing about wavelet codecs (for video especially) is that they are easy to scale by decoding portions of the bit stream. This is really useful for streaming as you can vary the image quality without too much work.
Isn't range coding widely viewed as being a legitimate workaround for the arithmetic coding patents? I guess in a year or two the question will be irrelevant, as the last of those patents will expire soon.
But at its core, WebP hasn't been shown to be more than marginally better than JPEG, so why bother modifying your tool chain and work flow? You generally need something to be an order of magnitude better for people to bother.