I don't know about much about the concrete plans for the Tor Messenger and CONIKS but I'm quite familiar with the original CONIKS design. First of all: I’m sure no one would force you to give your "real" identity, you could for instance use large  identity provider which is rather difficult to compromise, at least for non-state actors (for example gmail and the pseudonym simplesmtptest123 ;-). Maybe, for the Tor messenger integration there will be/people might choose some other identity providers (with a stronger focus on privacy and more freedom to choose pseudonyms instead of real names). 

If an identity provider (one of the several "CONIKS servers") is compromised, the attacker is able to read the provider's local directory (containing public key of already registered providers), he would basically see a more or less ‘randomly' looking Merkle tree. Theoretically, the attacker would still need to know all the user real-names beforehand to (for instance) query for their public keys. (This is achieved using the following "crypto-tricks": identities are stored at a private “index" in the tree; computed using a verifiable unpredictable function from a cryptographic commitment/hash of the username instead from the username itself). Of course one would also need to make sure that the stored public-key material (in the leaf-nodes) is pruned from user identifying data (like an identity in GPG); otherwise the attacker could guess the identities from that information. 

Also, in general, the attacker won’t be able to see that you used Tor Messenger from the mere fact that you use a certain identity provider, even if he still could recompute your user-name from the directory.

Hope that helps?

Ismail