The order $O_n(σ)$ of a permutation $σ$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $σ$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdös and Turán who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We extend this result to the so-called \textit{generalized Ewens measure} in a previous paper. In this paper, we establish a local limit theorem as well as, under some extra moment condition, a precise large deviations estimate. These properties are new even for the uniform measure. Furthermore, we provide precise large deviations estimates for random permutations with polynomial cycle weights.