"quickly drafted" is correct and usable in written English.
You can use it when describing something that was prepared quickly and in a hurry. For example, "The team quickly drafted a proposal for the new project."
At least two police casualties were reported and reinforcements were quickly drafted to the flash point.
The medics were quickly drafted into the effort to save Ms. Yousafzai's life.
Their congressional supporters quickly drafted legislation to set that standard into law.
The petition was quickly drafted, begging Mr. Fuentes to ignore negative comments that were "the product of nervousness and desperation".
He was quickly drafted, in 1959, into the top-secret space-flight program promoted by Nikita S. Khrushchev.
When asked to recommend another restaurant, Linda Rasberg, the headwaiter at the Michelin-starred Fond, quickly drafted an annotated list of eight suggestions.
The breaking news tool provides only the core functionality necessary to find a story and quickly draft the alert headline.
"I'm not sure how we could have a closed session without the accused until there is a ruling on that issue," said Colonel Pohl, who asked the lawyers to quickly draft briefs on the question.
In forensics we'd tackle this issue by quickly drafting a structure on a notecard to support our main point — often an introduction, two or three supporting points, and a conclusion.
But then again, many Google Docs users already use the service to quickly draft something.
As you type your email, EasyEmail digests the context of the email you're responding to, and suggests potential phrases or sentences to quickly draft your email.
