If you're at the stage of verbing your own nouns, you should be able to dispense with the "How to verb?" construction, which does not exist in proper English.
To your question:
1. ”The customary notice period was waived, and almost immediately he left the country to take up a new appointment overseas.”
2. ”The customary notice period was waived and, almost immediately, he left the country to take up a new appointment overseas.”
In 1. the comma is optional, and signifies a pause.
In 2. The twin commas surround the parenthesised adverbial phrase.
Your editor's version is not correct. If you parenthesise a phrase like this, the sentence must make sense even if you delete the phrase.