Context. Most massive stars reside in multiple systems that will interact over the course of their lifetime. This has important consequences on their future evolution and their end-of-life products. Classical Wolf-Rayet (WR) stars represent the final end stages of stellar evolution at the upper-mass end. While their observed multiplicity fraction is reported to be similar to 0.4 in the Galaxy, their intrinsic multiplicity properties and the distributions of their orbital parameters remain insufficiently constrained to provide a reliable anchor to compare to evolutionary predictions. Aims. As part of a homogeneous, magnitude-limited (V <= 12) spectroscopic survey of northern Galactic WR stars, this paper aims to establish the observed and intrinsic multiplicity properties of the early-type nitrogen-rich WR population (WNE), including estimates of the multiplicity fraction and the shape of their orbital period distribution. Additionally, we compare these with the properties of the carbon-rich WR population (WC) stars obtained in the first paper of this series. Methods. We obtained high-resolution spectroscopic time series of the complete magnitude-limited sample of 16 WNE stars observable with the 1.2 m Mercator telescope at La Palma, typically providing a time base of about two to eight years. We measured relative radial velocities (RVs) using cross-correlation and used RV variations to flag binary candidates. Using an updated Monte Carlo method with a Bayesian framework, we calculated the three-dimensional likelihood for the intrinsic binary fraction (f(int)(WNE)), the maximum period (log P-max), and the power-law index for the period distribution (pi) for the WNE population with P-min fixed at 1 d. We also used this updated method to re-derive multiplicity parameters for the Galactic WC population. Results. Adopting a peak-to-peak RV variability threshold of 50 km s(-1) as a criterion, we classify 7 of the 16 targets as binaries. This results in an observed multiplicity fraction (f(obs)(WNE)) of 0.44 +/- 0.12. Assuming flat priors, we derive the best-fit multiplicity properties f(int)(WNE) = 0.56(-0.15)(+0.20), log P-max = 4.60(-0.77)(+0.40), and pi = -0.30(-0.53)(+0.55) for the parent WNE population. We explored different mass-ratio distributions and note that they did not change our results significantly. For the Galactic WC population from Paper I, we re-derive f(int)(WC) = 0.96(-0.22)(+0.04), log P-min = 0.75(-0.60)(+0.26), log P-max = 4.00(-0.34)(+0.42), and pi = 1.90(-1.25)(+1.26). Conclusions. The derived multiplicity parameters for the WNE population are quite similar to those derived for main-sequence O binaries but differ from those of the WC population. The significant shift in the WC period distribution towards longer periods is too large to be explained via expansion of the orbit due to stellar winds, and we discuss possible implications of our results. Analysis of the WNL population and further investigation of various evolutionary scenarios is required to connect the different evolutionary phases of stars at the upper-mass end.
