urgently/on an urgent basis

Urgent: please edit this article.

So neither sentence of mine is used by natives. Oh dear!

"Please edit the article as soon as possible" sounds more natural and conveys the same meaning.

"Please edit the article ASAP" puts even more emphasis on urgency...

"Urgently" doesn't mean "quickly." It's more along the lines of "pressing."

There's an urgency for the article to be edited.

What your sentences convey is a sense that the articles themselves should be edited along the lines of urgency or pressingness.

Tang, again context. Either might mean to work up an agitated state before editing

The second might mean, as an extreme example, that you have been allowed a number of different bases upon which to base your editing but in this particular instance you should employ urgency