I'm wondering whether it is possible to take any distro other than Ubuntu (and its derivatives) and put Unity/Ubuntu-HUD on top of it, just like you would for instance install LXDE onto any distro that came with a different desktop that you didn't like?

Yes, just install Unity not the full ubuntu-desktop meta package.

I have done apt-get install ubuntu-desktop on my machine running Bodhi Linux, but practically got it transformed into Ubuntu, that install pulling all of the Ubuntu apps automatically.

Because you basically installed the default configuration for Ubuntu. For example, Installing lxdeis not the same as Installing lubuntu-desktop.

What I wanted though was to use only the applications I had, in a different GUI. Is this possible, and if yes, how?

Just install unity with the suffix --no-install-recommends