回答
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The work significantly delayed
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because of the defects of the sewing machine in the curtain sewing process.
大幅に遅れるは、significantly delayed, 機械などの調子が悪い、は、defects of ... , 縫製の時は、sewing process, と表現します。
回答
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Due to the sewing machine being in a bad shape, the progress on sewing my curtains was delayed to a large extent.
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When the condition of a machine is bad, work cannot progress smoothly.
「カーテン縫製の時、ミシンの調子が悪くて作業が大幅に遅れた」は英語に訳すとこの用は自然です。
Due to the sewing machine being in a bad shape, the progress on sewing my curtains was delayed to a large extent.
ミシン＝ sewing machine
カーテン縫製= to sew curtains
調子が悪い＝ to be in bad shape; to be in a bad condition
大幅に遅れた= delayed to a large extent
機械＝ machine
作業= work
スムーズ＝ smoothly
「機械の調子が悪いと、作業がスムーズに進まない」は英語では「When the condition of a machine is bad, work cannot progress smoothly」になります。
回答
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There was a problem with the machine when we were sewing the curtains, so production has been (significantly) delayed.
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The machine broke down when we were sewing the curtains, so production has been (significantly) delayed.
英訳1：「作業が遅れた」は「製品が遅れた」と言い換えることもできるので、production has been delayed を使いました。by ~ days/weeks/months「~日/~週/~月」を添えると、具体的に伝わりますね。
英訳2：「調子が悪い」機械について話す場合、there was a problem「問題があった」や、 broke down「壊れた」などが一般的な言い回しです。
日本 
アメリカ合衆国 
