I used "Chat Heads" before on version 1.19.2 on Fabric (with OptiFabric and Optifine) and everything worked. Now I tried on 1.20.1 in the same way and I see the head icons only in TAB. Besides everything I tried to change in the settings using the Mod Menu, nothing seems to work. I even downloaded the Cloth Config and... I don't see any results. It worked before even with the Chat Patches mod.

Using Fabric on 1.20.1 with Optifine
The Chat Heads mod is 0.10.16
Cloth Config is 11.1.106

for me it doesn't even load on 1.20.1, showing error:

Mod 'Chat Heads' (chat_heads) 0.10.16 requires any 1.19.x version of 'Minecraft' (minecraft), but only the wrong version is present: 1.20.1!

As with #54, Optifine is not supported.
We can't do anything about it since it is closed source, doesn't matter if it worked before.
Use Sodium and co instead. Here's a good list of open source alternatives.

I did just test Chat Patches on 1.20 and it seems to work fine with Chat Heads, so it's definitely Optifine.

Also, @KruASe76 you're not using the right Chat Heads version. You have a 1.19 version when you need the 1.20 version.