Yes, AEAD constructions always authenticate the nonce. If they didn't then the integrity and authentication of the resulting plaintext after decryption would not be ensured.
You can see the definition in RFC 5116: An Interface and Algorithms for Authenticated Encryption, section 2.1. Authenticated Encryption:
The nonce is authenticated internally to the algorithm, and it is not
necessary to include it in the AD input. The nonce MAY be included
in P or A if it is convenient to the application.
Not all AEAD ciphers may necessarily keep to the letter of this RFC (the inclusion of the authentication tag in the ciphertext is debatable if you ask me - I consider it a separate field). However, I haven't seen any AEAD cipher that doesn't authenticate the nonce by design.
You're writing that
With a more "traditional" composition, like AES-CBC-then-HMAC, the nonce is obviously authenticated.
This is obviously not true. Yes, it is possible to design such a scheme and any competent cryptographer would not for get to authenticate the IV. There are even tries to standardize such a scheme, see e.g this scheme based on CBC with HMAC-SHA-1.
Less competent protocol designers are however free to shoot themselves in the foot when designing or implementing such a scheme. When using a well designed / reviewed / tested library and an AEAD scheme such mistakes are avoided.