Without going to heroics, there has been no way for the average user to use the built-in Java viewer for a number of years. By 2017 no real browser supported the NPAPI java plugin, and by 2020 the plugin was removed from Java.
Instead of including a web-viewer in the format of a Java applet that hardly anyone can use, perhaps some consideration can be given to including noVNC? It's available now, it works, it uses HMTL5 & javascript, is compatible with every major browser, and it should be basically a drop-in replacement for the current java applet inside UVNC. It should be pretty simple to deploy with UVNC.
It also has the benefit of being a separately maintained project with its own community, which offloads responsibility for maintenance from the UVNC project.