The 15N isotopologue abundance ratio measured today in different bodies of the solar system is thought to be connected to 15N-fractionation effects that would have occured in the protosolar nebula. The present study aims at putting constraints on the degree of 15N-fractionation that occurs during the prestellar phase, through observations of D, 13C and 15N-substituted isotopologues towards B1b. Both molecules from the nitrogen hydride family, i.e. N2H+ and NH3, and from the nitrile family, i.e. HCN, HNC and CN, are considered in the analysis. As a first step, we model the continuum emission in order to derive the physical structure of the cloud, i.e. gas temperature and H2 density. These parameters are subsequently used as an input in a non-local radiative transfer model to infer the radial abundances profiles of the various molecules. Our modeling shows that all the molecules are affected by depletion onto dust grains, in the region that encompasses the B1-bS and B1-bN cores. While high levels of deuterium fractionation are derived, we conclude that no fractionation occurs in the case of the nitrogen chemistry. Independently of the chemical family, the molecular abundances are consistent with 14N/15N~300, a value representative of the elemental atomic abundances of the parental gas. The inefficiency of the 15N-fractionation effects in the B1b region can be linked to the relatively high gas temperature ~17K which is representative of the innermost part of the cloud. Since this region shows signs of depletion onto dust grains, we can not exclude the possibility that the molecules were previously enriched in 15N, earlier in the B1b history, and that such an enrichment could have been incorporated into the ice mantles. It is thus necessary to repeat this kind of study in colder sources to test such a possibility.