What kind of attack could one perform if you generate private keys using SHA3(master_key || nonce)?
KDFs

What you are trying to describe are different implementations of a key based key derivation function (KBKDF) which in turn is a specific form of key derivation function (KDF) that uses a key (and not a password).

The attacker would not be able to reverse the hash when using SHA3(master_key || nonce). The reversal could not even happen for SHA-1; the fact that SHA-1 is broken for collision resistance does not matter for this construction. HMAC is more appropriate because a keyed hash / PRF has better cryptographic properties than a hash.

$H(k | c)$ is actually a known KDF called KDF1 or 2 (those just differ in initial counter value $c$). As described here it is probably still secure. However, if you want real, standardized, proven security I'd probably recommend HKDF using SHA-512 as underlying hash. That should be secure until SHA-512 is broken to the extreme.

On HMAC

HMAC protects the underlying hash function against certain attacks. It should always be secure if a large enough key is used and the underlying hash is sufficiently secure - but that's even the case for MD5. HMAC also alleviates length extension attacks for the underlying hash. However, for SHA-3 there should be no length extension attacks possible, so SHA3(master_key || nonce) would not be affected by length extension attacks anyway.

HMAC is rather overkill for SHA-3 as it hashes the key twice after applying two different masks, while the protection it offers is inconsequential to SHA-3. So HMAC-SHA3 could considered an option because the HMAC construction is secure with any sufficiently secure hash. The more efficient KMAC construction is described in SP 800-185 and could be used instead - if it is available in the programming environment anyway.

TL;DR

Now that you know that you need a KBKDF, I suppose it is a good idea to use a well described one. I'd strongly recommend HKDF or HKDF-expand. NIST SP 800-108 has a few more KBKDF's, but NIST standardized HKDF afterwards for good reasons.