The Vulkan RenderSystem backport from Ogre-next, now has landed in the master branch and will be available with Ogre 13.2. See the screenshot below for the SampleBrowser running on Vulkan

The code was simplified during backporting, which shows by the size reduction from ~33k loc in Ogre-Next to ~9k loc that are now in Ogre.

The current implementation pretends to have Fixed Function capabilities, which allows operating with one default shader – similarly to what I did for Metal. This shader only supports using a single 2D texture without lighting. E.g. vertex color is not supported. This is why the text is white instead of black in the screenshot above.
Nevertheless, it already runs on Linux, Windows and Android.

Proper lighting and texturing support, will require some adaptations to the GLSL writer in the RTSS, as Vulkan GLSL is slightly different to OpenGL GLSL. This, and the other currently missing features will hopefully come together during the 13.x development cycle. If you are particularly keen on using Vulkan, consider giving a hand.
Right now, the main goal is to get Vulkan feature-complete first, so dont expect it to outperform any of the other RenderSystems. Due to being incomplete, the Vulkan RenderSystem is tagged EXPERIMENTAL.