We study the effect of the intrinsic (Rashba or Dresselhaus) spin-orbit interaction in superconductor-nanowiresuperconductor (SNS) weak links in the presence of a spin-splitting field that can result either from an intrinsic exchange field or the Zeeman effect of an applied field. We solve the full nonlinear Usadel equations numerically [The code used for calculating the results in this paper is available in https://github.com/wompo/Usadel-fornanowires] and analyze the resulting supercurrent through the weak link and the behavior of the density of states in the center of the wire. We point out how the presence of the spin-orbit interaction gives rise to a long-range spin triplet supercurrent, which remains finite even in the limit of very large exchange fields. In particular, we show how rotating the field leads to a sequence of transitions between the 0 and π states as a function of the angle between the exchange field and the spin-orbit field. Simultaneously, the triplet pairing leads to a zero-energy peak in the density of states. We proceed by solving the linearized Usadel equations, showing the correspondence to the solutions of the full equations and detail the emergence of the long-range supercurrent components. Our studies are relevant for ongoing investigations of supercurrent in semiconductor nanowires in the limit of several channels and in the presence of disorder.