I was able to fix it by loosely following this page, though it has some errors.

What worked:
Install the font:

sudo apt install fonts-noto-color-emoji

Create ~/.config/fontconfig/fonts.conf with the following (note that the page above is missing <fontconfig>)

<?xml version="1.0" encoding="UTF-8"?><!DOCTYPE fontconfig SYSTEM "fonts.dtd">
<fontconfig>
 <match>
  <test name="family"><string>sans-serif</string></test>
  <edit name="family" mode="prepend" binding="strong">
   <string>Noto Color Emoji</string>
  </edit>
 </match>

 <match>
  <test name="family"><string>serif</string></test>
  <edit name="family" mode="prepend" binding="strong">
   <string>Noto Color Emoji</string>
  </edit>
 </match>

 <match>
  <test name="family"><string>Apple Color Emoji</string></test>
  <edit name="family" mode="prepend" binding="strong">
   <string>Noto Color Emoji</string>
  </edit>
 </match>
</fontconfig>

(Note: if fontconfig exists [it didn't on my install] then more finesse will be required here)

Flush the font cache with:

fc-cache -f -v

Reboot or load the page and it shows the proper cloud emoji:

Hey, if you can post your full working ~/.config/fontconfig/fonts.conf I could add a fix for this.

If I may make a suggestion, OpenMoji recently got a full update to the Unicode 15 standard:

https://github.com/hfg-gmuend/openmoji/releases/tag/15.0.0

I personally think it would be handy to implement it as the default Emoji setting that comes built-in with the system.
Could double as a fall-back in case of further errors like these.

@SecularSteve Thanks for the tip:
Related, this is what we do now: #378

Nightly build are working again.

@jkriegshauser If I make a build with a fix for it could you test it?

@guysoft Sure

didn't forget, I want to fix the username bug first (#525)

Didn't forget, but I am really really swamped.