In general, it varies. There are cases where the convergence is quite fast (for example in the case where the system is mixing, and say in the presence of spectral gap, think of Bernoulli system or say geodesic flow on $PSL_{2}(\mathbb{R})/\Gamma$).
On the other hand, the convergence can be rather slow in Kronecker systems (hence also in the Kronecker factor of your system, which will appear in the Hilbert space $L^{2}(\mu)$ unless your system is weak-mixing). The reason is simple, you need to estimate the exponential sum $\frac{1}{N}\sum_{n=0}^{N-1}e_{\alpha}(n)$ where $e_{\alpha}(x)=exp(2\pi i \alpha x)$ and $\alpha$ is the corresponding eigenvalue. In the case that $\alpha$ is Liouvillian the decay rate of this sum can be rather bad. Somehow the most precise results appear in the famous Green-Tao paper about nilflows - http://arxiv.org/abs/0709.3562 .