We present results from radiation non-ideal magnetohydrodynamics (MHD) calculations that follow the collapse of rotating, magnetised, molecular cloud cores to stellar densities. These are the first such calculations to include all three non-ideal effects: ambipolar diffusion, Ohmic resistivity and the Hall effect. We employ an ionisation model in which cosmic ray ionisation dominates at low temperatures and thermal ionisation takes over at high temperatures. We explore the effects of varying the cosmic ray ionisation rate from $zeta_text{cr}= 10^{-10}$ to $10^{-16}$ s$^{-1}$. Models with ionisation rates $gtrsim 10^{-12}$ s$^{-1}$ produce results that are indistinguishable from ideal MHD. Decreasing the cosmic ray ionisation rate extends the lifetime of the first hydrostatic core up to a factor of two, but the lifetimes are still substantially shorter than those obtained without magnetic fields. Outflows from the first hydrostatic core phase are launched in all models, but the outflows become broader and slower as the ionisation rate is reduced. The outflow morphology following stellar core formation is complex and strongly dependent on the cosmic ray ionisation rate. Calculations with high ionisation rates quickly produce a fast (~14km s$^{-1}$) bipolar outflow that is distinct from the first core outflow, but with the lowest ionisation rate a slower (~3-4 km s$^{-1}$) conical outflow develops gradually and seamlessly merges into the first core outflow.
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