I started with a Chapterhouse UK proofreading distance course. But perhaps that should be 'ended with'!
Before that, I had completed a BA(Hons) degree in Creative Writing as part of a Creative Arts, Combined Studies degree course. During this, I learned to write and craft my own writing which, although I had already had articles published locally, led to articles and features, stories, radio, etc., being published nationally.
After finishing my degree, I gained a Further and Adult Teaching Certificate.
Then I taught writing-related courses part time, first at local night school classes, then later at a University. During this, I would have to 'mark' the work of my students - pointing out the errors, suggesting improvements, etc., - basically doing what an editor does.
Then came the Chapterhouse course, which I did mainly out of curiosity!
Then I taught basic proofreading at the university as well as the other courses, and helped other people to become proofreaders.
So that was the way I became an editor and proofreader. It took about 6 years altogether. And I should also mention that I went to an English Grammar school, where we studied English grammar to the nth degree. I hated it at the time!
I suppose what I'm trying to say here is that you will need some sort of qualifications to prove you can actually do the job.
Edited to add: You don't need all the qualifications I have to become a good editor or proofreader, but you do need to prove that you can do the job well! OP - you have the right background, you just need some specialist training, whether that be a correspondence course or night school classes.