Keep the explanation brief and professional. You can say that an unavoidable work commitment has arisen without giving the recruiter a detailed story.

Recruiters mainly need to know that the conflict is legitimate, that you remain interested, and what alternative times you can make. Too much explanation can create unnecessary complexity.

A concise message can say that an unavoidable client or work commitment now conflicts with the scheduled interview, apologize for the inconvenience, reaffirm your interest, and offer several alternative times.

Contact recruiting as soon as you know about the conflict. Provide two or three realistic alternatives and avoid repeated changes unless absolutely necessary.

Keep it short. Recruiter doesn't need the full workplace drama.

Giving alternative times right away makes the email much easier.