I've been preparing for an interview, and I've seen both of these phrases used when introducing yourself and thanking someone for their time or when support is being given. For example:
お忙しいなかお時間をとって いただきありがとうございます
お忙しいところ大変恐縮ではございますが、何卒よろしくお願いいたします
Is there a nuanced difference between using ところ and なか in this kind of situation, or are they generally interchangeable? As well, what is the direct translation of ところ・なか?
Those two are interchangeable. We don't feel any difference if they follow 忙しい. People say differently just by habit. If you need some source => https://mayonez.jp/topic/1036754 (Note we don't have official recieved one, some manner instructor have slightly different policy. Also it depends on region, industry, company who belongs to)
what is the direct translation of ところ・なか?
if you want direct translation and make them different, it's close to
忙しい中(なか) = Of/In busy time(days).
忙しい所(ところ) = Of/In your busy situation.
So whole sentences become (might be weird in English)
お忙しいなかお時間をとって いただきありがとうございます
Thank you for taking time in your busy schedule.
お忙しいところ大変恐縮ではございますが、何卒よろしくお願いいたします
Although you are in busy situation, let us ask your favor for this.
