。、。
I'm sorry for the late reply. I've been swamped with work lately.
Literal
Reply [subject-が] was-late apologies-was. Lately, work [by-に] overwhelmed-being-humbly.
Two sentences of polished business apology. 申し訳ありませんでした uses the past tense to acknowledge a completed failing. このところ means 'recently/lately.' 忙殺されておりまして layers the passive (される), humble ongoing form (おる for いる), and the formal connective まして — explaining the reason without making excuses. The passive frames the busyness as something imposed on the speaker.