I'm so lucky my day off from work happened to be on the same day as New Year's Day.
I'm so lucky my vacation day happened to be one the same day as New Year's Day.
I'm so lucky my vacation day unexpectedly became on the same day as New Year's Day.
「今回はたまたま仕事の休みと元旦が重なったのでラッキー」は英語に訳すならこのようになります。
I'm so lucky my day off from work happened to be on the same day as New Year's Day.
I'm so lucky my vacation day happened to be one the same day as New Year's Day.
I'm so lucky my vacation day unexpectedly became on the same day as New Year's Day.
「たまたま」→ unexpectedly, happened to be
「仕事の休み」→ vacation day(s), day(s) off
「元旦」→ New Year's Day
何かの予定などが「重なった」時に on the same day (同じ日)や at the same time (同じ時)と言いますが、この場合は同じ日なので、 on the same day と言います。
ご参考までに。