The Digilent Petalinux projects here all include HDMI output:
They use the Xilinx DRM driver, with an additional "slave encoder" driver that adds support for Digilent video output pipelines (both VGA and HDMI work with the driver). The linux-digilent tree includes device-tree binding Documentation for the slave encoder driver. Really, the driver just adds support to the Xilinx DRM for any sink that uses the stream2video IP core, so it is possible that it gets replaced by work that Xilinx does in the future (Last I spoke with them they weren't interested in accepting upstreamed work).
I wrote the driver, feel free to ask questions. I'm not familiar with how video output in Xillinux works, I never used it. None of our designs use the Xylon cores, we are trying to keep things as free and open source as possible.
What you are doing shouldn't take much work, the hard part will probably be finding the ubuntu image you want to use. You could also try to build one yourself using Petalinux (which is mostly a wrapper for Yocto).