There was a problem with the machine when we were sewing the curtains, so production has been (significantly) delayed.
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。
Due to the sewing machine being in a bad shape, the progress on sewing my curtains was delayed to a large extent.
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。
日本 
アメリカ合衆国 
