QA automation is coding using specific library, usually Se Webdriver. If your goal is to advance toward management positions, advanced coding skills are almost irrelevant (basic coding skills you already have might be enough).
Dare I say, not being CompSci graduate might be even a positive, because CompSci graduates tend to be introverts, and management position requires more focus on communication skills (usually not a strong point of introverts).
If your coding skills are OK, and you want to be a QA manager, I would suggest (when considering T profiled skillsT profiled skills from @FDM answer) to make your communication skills the deep part, and cover the shallow part just enough: Become OK in networking, system administration, database administration and design, have intro level knowledge of most tools your QA uses. So you can understand the concerns of the different members of your team and negotiate between different priorities, without micromanaging (let them own the technical side of the problem).
Hard part is how to maintain adequate sharpness of your skills in all those areas without spending too much time in any of them, when so much of your time would be spent in meetings... :-(