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This paper contributes to the current studies on regularity properties ofnoncommutative distributions in free probability theory. More precisely, weconsider evaluations of selfadjoint noncommutative polynomials innoncommutative random variables that have finite non-microstates free Fisherinformation, highlighting the special case of Lipschitz conjugate variables.For the first time in this generality, it is shown that the analyticdistributions of those evaluations have H\"older continuous cumulativedistribution functions with an explicit H\"older exponent that depends only onthe degree of the considered polynomial. For linear polynomials, we reach inthe case of finite non-microstates free Fisher information the optimal H\"olderexponent $\frac{2}{3}$, and get Lipschitz continuity in the case of Lipschitzconjugate variables. In particular, our results guarantee that such polynomialevaluations have finite logarithmic energy and thus finite (non-microstates)free entropy, which partially settles a conjecture of Charlesworth andShlyakhtenko [CS16].  We further provide a very general criterion that gives for weakapproximations of measures having H\"older continuous cumulative distributionfunctions explicit rates of convergence in terms of the Kolmogorov distance.  Finally, we combine these results to study the asymptotic eigenvaluedistributions of polynomials in GUEs or matrices with more general Gibbs laws.For Gibbs laws, this extends the corresponding result obtained in [GS09] fromconvergence in distribution to convergence in Kolmogorov distance; in the GUEcase, we even provide explicit rates, which quantify results of [HT05,HST06] interms of the Kolmogorov distance.

H\"older Continuity of Cumulative Distribution Functions for Noncommutative Polynomials under Finite Free Fisher Information