Many servers allow users to create nicknames for themselves (like ~Poly), but these do not register by this mod. Would it be possible to allow users to define aliases for player names in the mod config?

k, backport to 1.19.2 / 1.19.1 is up!
Well, kinda, Modrinth seems to have technical trouble right now.

If anyone reads this and wants a backport for earlier versions, just ask!

Do these nicknames show in the player list when pressing tab or is this purely in the messages that get sent?
If it's the former, this sounds like a bug or an issue with Chat Heads' settings which "Heuristic only" sender detection could fix - give that a try if you haven't.
If it's the latter, we probably need something like aliases like you suggest.

Here is a scenario on a server which does not provide sender UUIDs.
BUSHI_YAMMAMOTO is nicknamed as ~Yamamoto, but their skin does not show up in chat with heuristic detection.
.Owlet1124 is a bedrock player logged on through GeyserMC, but their skin is available to the client as show in the image. Their nickname is ~Owl.

k, in this case I think I can push my branch - already wrote an implementation last week.

I still find it so weird that they made nicknames separate from display and profile names - display names are the "vanilla" way to nickname players (vanilla itself doesn't directly use that functionality; they literally made it so that modded servers could rename players), but oh well.

Oh right, I only wrote the implementation for 1.19.3, is that fine or do you need a backport?

P.S.
Just released 0.9.0 for 1.19.3

Nice work!
I have not had the opportunity to update to 1.19.3 yet as not all mods I depend on have released for the version yet, so I am still on 1.19.2. A back-port would be very kind.