The chemical recycling of cellulosic fibres may represent a next-generation fibre–fibre recycling system for cotton textiles, though remaining challenges include how to accommodate fibre blends, dyes, wrinkle-free finishes, and other impurities from finishing. These challenges may disrupt the regeneration process steps and reduce the fibre quality. This study examines the impact on regenerated viscose fibre properties of a novel alkaline/acid bleaching sequence to strip reactive dyes and dimethyloldihydroxyethyleneureas (DMDHEU) wrinkle-free finish from cotton textiles. Potentially, such a bleaching sequence could advantageously be integrated into the viscose process, reducing the costs and environmental impact of the product. The study investigates the spinning performance and mechanical properties (e.g., tenacity and elongation) of the regenerated viscose fibres. The alkaline/acid bleaching sequence was found to strip the reactive dye and DMDHEU wrinkle-free finish from the cotton fabric, so the resulting pulp could successfully be spun into viscose fibres, though the mechanical properties of these fibres were worse than those of commercial viscose fibres. This study finds that reactive dyes and DMDHEU wrinkle-free finish affect the viscose dope quality and the regeneration performance. The results might lead to progress in overcoming quality challenges in cellulosic chemical recycling.