We extend our initial study of diffusive nuclear burning (DNB) for neutron stars (NSs) with hydrogen (H) atmospheres and an underlying layer of proton-capturing nuclei. Our initial study showed that DNB can alter the photospheric abundance of hydrogen on surprisingly short timescales (10~2-10~4 yr). Significant composition evolution impacts the radiated thermal spectrum from the NS as well as its overall cooling rate. In this paper, we consider the case when the rate-limiting step for the H consumption is diffusion to the burning layer rather than the local nuclear timescale. This is relevant for NSs with surface temperatures in excess of 10~6 K, such as young (<10~5 yr) radio pulsars and accreting NSs in quiescence. When downward diffusion is the limiting rate in DNB, the rate of H consumption is suppressed by 1-2 orders of magnitude compared to a DNB estimate that assumes diffusive equilibrium. In order to apply our ongoing study to young neutron stars, we also include the important effects of strong magnetic fields (B ～ 10~(12) G). In this initial study of magnetic modifications to DNB, we find that the H-burning time is lengthened by 2-3 orders of magnitude for a 10~(12) G field. However, even for NSs with dipole field strengths of 10~(12) G, we find that all of the H can be burned before the pulsar reaches an age of ～10~5 yr, thus potentially revealing the underlying proton-capturing elements. Finally, we conclude by providing an overview of what can be learned about fallback and pulsar winds from measuring the surface composition of a young NS.
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