When you’ve been too busy to reply to someone’s email:

Sorry! I haven’t been attending my emails. I have been,and still am, snowed under. [Give brief details, e.g., I’ve been on-call all week at the hospital. . .)

When someone sends a follow-up email becasue you haven’t replied to their first email:

Sorry! I have been rude– but not intentionally — just haven’t got to it yet. (give brief reason for your busyness.) It’s with the other 100 emails to attend to now I’ve [got this talk out of the way–or whatever].