We present Born-Oppenheimer (BO) and Renner-Teller (RT) time dependent quantum dynamics studies of the reactions O((3)P) + H2(+)(X(2)Σg(+)) → OH(+)(X(3)Σ(-)) + H((2)S) and OH(X(2)Π) + H(+). We consider the OH2(+) X[combining tilde](2)A'' and Ã(2)A' electronic states that correlate with a linear (2)Π species. The electronic angular momenta operators L[combining circumflex] and L[combining circumflex](2) are considered in nonadiabatic coupled-channel calculations, where the associated RT effects are due to diagonal V(RT) potentials that add up to the PESs and to off-diagonal C(RT) couplings between the potential energy surfaces (PESs). Initial-state-resolved reaction probabilities PI, integral cross sections σI, and rate constants kI are obtained using recent ab initio PESs and couplings and the real wavepacket formalism. Because the PESs are strongly attractive, PI have no threshold energy and are large, σI decrease with collision energy, and kI depend little on the temperature. The X[combining tilde](2)A'' PES is up to three times more reactive than the Ã(2)A' PES and H2(+) rotational effects (j0 = 0, 1) are negligible. The diagonal V(RT) potentials are strongly repulsive at the collinearity and nearly halve all low-energy observables with respect to the BO ones. The off-diagonal C(RT) couplings are important at low partial waves, where they mix the X[combining tilde](2)A'' and Ã(2)A' states up to ∼20%. However, V(RT) effects predominate over the C(RT) ones that change at most by ∼19% the BO values of σI and kI. The reaction O((3)P) + H2(+)(X(2)Σg(+)) → OH(+)(X(3)Σ(-)) + H((2)S) is probably one of the most reactive atom + diatom collisions because its RT rate constant at room temperature is equal to 2.26 × 10(-10) cm(3) s(-1). Within the BO approximation, the present results agree rather well with recent quasiclassical and centrifugal-sudden data using the same PESs.