Topological semimetals have been at the forefront of experimental and theoretical attention in condensed-matter physics. Among these, recently discovered Weyl semimetals have a dispersion described by a three-dimensional Dirac cone, which is at the root of exotic physics such as the chiral anomaly in magnetotransport. In a time-reversal symmetric (TRS) Weyl semimetal film, the confinement gap gives the quasiparticles a mass, while TRS is preserved by having an even number of valleys with opposite masses. The film can be tuned through a topological phase transition by a gate electric field. In this work, we present a theoretical study of the quantum corrections to the conductivity of a topological semimetal thin film, which is governed by the complex interplay of the chiral band structure, mass term, and scalar and spin-orbit scattering. We study scalar and spin-orbit scattering mechanisms on the same footing, demonstrating that they have a strong qualitative and quantitative impact on the conductivity correction. We show that, due to the spin structure of the matrix Green's functions, terms linear in the extrinsic spin-orbit scattering are present in the Bloch and momentum relaxation times, whereas previous works had identified corrections starting from the second order. In the limit of small quasiparticle mass, the terms linear in the impurity spin-orbit coupling lead to a potentially observable density dependence in the weak antilocalization correction. Moreover, when the mass term is around 30% of the linear Dirac terms, the system is in the unitary symmetry class with zero quantum correction, and switching the extrinsic spin-orbit scattering drives the system to the weak antilocalization. We discuss the crossover between the weak localization and weak antilocalization regimes in terms of the singlet and triplet Cooperon channels, and analyze this transition as a function of the mass and spin-orbit scattering strength. Experimental schemes to detect this transition are discussed.