"多轴纺织机" can be translated into English as "multi-spindle textile machine" or "multi-axis textile loom" (if specifically referring to a loom).
"Multi-spindle" emphasizes the multiple spinning units or spindles in the machine.
"Multi-axis" highlights the multiple axes or directions of movement or operation, which might be more relevant if the machine has complex motion patterns.
In most contexts, especially when referring to a general-purpose textile machine with multiple spindles, "multi-spindle textile machine" would be a suitable and commonly understood term. If the context specifically involves a loom (a type of textile machine for weaving), then "multi-axis textile loom" could also be appropriate, though "multi-spindle" might still be used depending on the exact design and function of the loom.