Low energy supersymmetry with bilinear breaking of R parity leads to a weak-scale seesaw mechanism for the atmospheric neutrino scale and a radiative mechanism for the solar neutrino scale. The model has striking implications for collider searches of supersymmetric particles. Assuming that the lightest SUSY particle is the lightest neutralino we demonstrate that (i) the neutralino decays inside the detector even for tiny neutrino masses and (ii) measurements of the neutrino mixing angles lead to predictions for the ratios of various neutralino decay branching ratios implying an independent test of neutrino physics at future colliders, such as the CERN Large Hadron Collider or a Linear Collider. We study the lightest neutralino decay branching ratio predictions taking into account present supersymmetric particle mass limits as well as restrictions coming from neutrino physics, with emphasis on the solar and atmospheric neutrino anomalies.