There is growing interest in how the climate would change under net zero carbon dioxide emissions pathways as many nations aim to reach net zero in coming decades. In today's rapidly warming world, many changes in the climate are detectable, even in the presence of internal variability, but whether climate changes under net zero would be detectable is less well understood. Here, we use a bespoke set of 1000-year-long net zero carbon dioxide emissions simulations branching from different points in the 21st century to examine the detectability of large-scale and local climate changes as time passes under net zero emissions. We find that many changes under net zero become detectable within centuries. While local changes and changes in extremes are more challenging to detect, warming in the Southern Hemisphere and cooling in the Northern Hemisphere becomes detectable at many locations within a few centuries under net zero emissions. We also study detectability of differences in climate indices due to delays in achieving emissions cessation. We find that for global mean surface temperature and other large-scale indices, such as Antarctic and Arctic sea ice extent, the effects of even a five-year delay in emissions cessation are detectable. Short delays in emissions cessation result in significantly different local temperatures for most of the planet, and most of the global population. The long simulations used here help with identifying local climate change signals. Multi-model frameworks will be useful to examine confidence in these changes and ultimately improve understanding of post-net zero climate changes.