We do it every time we go on vacation or take a sick day. We put up an out-of-office (OOO) message with the date of our return, a colleague’s contact information for urgent needs, and maybe even some details about the destination of our long-awaited vacation.
I think the problem is that “at your earliest convenience” is a formulaic convention that uses explicit, almost exaggerated politeness to basically issue a stern direction, meaning “as soon as you possibly can”. When you turn it into “at my earliest convenience” it’s unclear if you mean “whenever it’s convenient for me to get to it” (what the words say) or “as soon as I possibly can” (what the meaning of the formulaic original is). Or else it sounds like you didn’t quite understand how “at your earliest convenience” works.

I hope this email finds you well. I’m out of the office right now but will get back to you as soon as possible. Expect a reply next Monday. For urgent matters, you can email or call [Name] at [email and phone number].
Hello! I’m going to be out of the office until [date]. If you need something before then, please contact [colleague] at [email] or [phone number] to receive assistance.Hello! Thank you for your email. I am out of the office right now but get back to you ASAP. You can expect a reply by [day] latest. For anything you need right away, you can contact [alternative contact option].