Apparently Ubuntu MATE is still on Xorg as of 24.04.
The quickest way to get Wayland (to the point of being able to run Waydroid) is to install weston (can be done via apt).
Launch Weston (needs to be done from the terminal as there is no launcher entry, but you can create one).
Run export WAYLAND_DISPLAY=wayland-1 (this is the display which Weston uses; Waydroid will read that variable; if not found, it will default to wayland-0 and exit if no display exists there).
Then run Waydroid normally.
Or run WAYLAND_DISPLAY=wayland-1 waydroid ARGS (replacing ARGS with first-launch for the first launch, show-full-ui on subsequent launches).
Caveat: Weston runs in a window, and Waydroid doesn’t seem to handle screen resize operations properly – if the window changes size, Waydroid will lose its display connection and disappear from the screen (although it remains running).