dyeing my hair took longer than expected so I was late for the lesson.
ご質問ありがとうございます。
「ヘアカラー」は動詞で使うとdye my hairとcolor my hairに訳せます。dyeの進行形はdyeingです。dying（死んでいる）と似てますので気を付けてください。
took longer than expectedは「思ったより時間がかかった」です。
ご参考になれば幸いです。
I got caught up in coloring my hair and ended up being late to the lesson.
「ヘアカラーに手間取ってレッスンに遅れてしまった」
- I got caught up in coloring my hair and ended up being late to the lesson.
- Coloring my hair took longer than expected so I ended up being late to the lesson.
自宅でヘアカラーをしていた。思ったより時間がかかってしまい、レッスンの時間に間に合わなかった。
- I was coloring my hair at home. It took longer than I thought I would and I was unable to make it to the lesson on time.
アメリカ合衆国 
