Upper bound for the length of non-interactive zero-knowledge proofs of knowledge
As Natanael pointed out, if you are willing to make very strong assumptions, SNARGs can be used to give proofs of constant size, independent of the size of the statement to be proven. However, these proof systems are in general not UC-compatible (see e.g. this paper). You have two natural alternatives:

1) use NIZKs with "almost optimal size", i.e., of size $w + \mathsf{poly}(k)$, where $w$ is the witness size, and $\mathsf{poly}(k)$ is a fixed polynomial in the security parameter. Such NIZKs exist from the LWE assumption (by combining this recent breakthrough and this paper), or from more exotic assumption. This is the generic, "heavy hammer" approach, if you really want to avoid thinking about how to concretely implement your NIZK.

2) Alternatively, you can use the right approach: look at the concrete statement you want to prove, write a NIZK for this statement, and look at its size.