Is there any functional encryption scheme whose key generation circuit is in NC1? To my best of knowledge, I have kown there exists randomized encoding scheme whose encoding circuit is in $NC^1$, fully homomorphic encryption scheme whose decryption circuit is in $NC^1$, and functional encryption scheme for $NC^1$ functions. Now I'm curious about is there any functional encryption scheme whose key generation circuit is in $NC^1$ ?
Attribute-Based Encryption schemes (ABE) are definitionally a sub-category of FE. Furthermore, non-monotonic ABEs (NM-ABE) can encode decisions from $NC^1$, so yes, there are FE-schemes that are restricted to $NC^1$.

There are both ciphertext-policy ABEs (CP-ABE), key-policy ABEs (KP-ABE) and mixtures of them (dual-policy, or DP-ABEs), so it is possible to embed the circuit of desired complexity into both ciphertext and key (or key generation).

The most general FE schemes can encode decisions from the whole of $NC$, so they are also able to encode circuits in $NC^1$. FE-schemes also exist in both KP- and CP-flavors.