Nuclear envelope reassembly during the final stages of each mitosis depends on disassembling spindle microtubules without disrupting chromosome separation. This process involves the transient recruitment of the ESCRT-III complex and spastin, a microtubule-severing AAA (ATPases associated with diverse cellular activities) mechanoenzyme, to late-anaphase chromosomes. However, dissecting mechanisms underlying these rapid processes, which can be completed within minutes, has been difficult. Here, we combine fast-acting chemical inhibitors with live-cell imaging and find that spindle microtubules, along with spastin activity, regulate the number and lifetimes of spastin foci at anaphase chromosomes. Unexpectedly, spastin inhibition impedes chromosome separation, but does not alter the anaphase localization dynamics of CHMP4B, an ESCRT-III protein, or increase γ-H2AX foci, a DNA damage marker. We show spastin inhibition increases the frequency of lamin-lined nuclear microtunnels that can include microtubules penetrating the nucleus. Our findings suggest failure to sever spindle microtubules impedes chromosome separation, yet reforming nuclear envelopes can topologically accommodate persistent microtubules ensuring nuclear DNA is not damaged or exposed to cytoplasm.