Stochastic analysis in engineering sciences takes into account the uncertainties that may exist and affect a certain physical system in an a priori unknown manner. As the design of structures gets increasingly complex over the years, the impact of those uncertainties onto the system response has to be studied in order to implement numerical procedures for virtual testing platforms. Braided composites are of special interest for the aerospace and the automotive industry, due to their excellent performance in terms of stiffness/strength-to-weight ratio, delamination resistance, impact properties etc. The complexity of such materials sets a computational challenge when it comes to robust and reliable simulations. Efficiency also plays an important role for probabilistic assessments since the response variability needs repetitive procedures in order to be calculated (e.g. Monte Carlo simulations). Hence, the aim of this work is to present an uncertainty quantification framework for braided composites simulation, dealing with the stochastic stiffness and strength prediction via numerical multiscale analysis. The numerical burden of Monte Carlo analysis is bypassed with various metamodeling techniques, such as Neural Networks, Polynomial Chaos expansion and Kriging modeling. Uncertainties accounting for material properties randomness, geometric randomness but also for random spatial variations caused by manufacturing aspects (e.g. fabric compaction during molding, jamming actions during braiding), are propagating through the scales to the final scatter of the mechanical properties of the macroscale. Results offer a perspective on the variability influence of the random parameters, an overview of the performance of several surrogate models and also highlight the importance of realistic uncertainty quantification. Furthermore, this work provides a useful guidance for uncertainty propagation assessment with advanced non-intrusive metamodeling techniques.