First post, hello!

I really want to get into game development, in design or visual art. I know a relevant degree is the easiest way to get into the industry, however, it just isn't possible for me. I started two undergrad degrees previously, one in English and the other in Digital Arts (where I gathered some programming experience). But because of health conditions, I couldn't complete my first degree or my second, which was my second chance. I had to drop out twice and now if I wanted to go back and try again at university, I cannot get funding for it because of my two previous attempts at uni. I couldn't afford £37,000 of tuition fees + living costs completely out of pocket unless I miraculously win the lottery.

So my question is, how could I go about breaking into the games industry without a degree? In this day and age with such high amounts of competition, is it even possible? Would QA be the way to go or should I try to make my own games through self-teaching?

On the very loosest terms, you can break into the game industry by simply putting up a game, even if it's free. So, no, a degree isn't required. However, a proper education, not necessarily a degree, but an education will get you very, very far. So, if you're unable to do college, then start teaching yourself and practice. In the end, You will have to be result driven. 

So where would you go from there? If it is art, then study art and animation. If it's programming ,study that. If it is writing, study writing. Etc. You will eventually find the resources you need from there because the knowledge of a skill is nothing new.  Just you have to have the knowledge and practice to build that foundation.

You aren't competing in a vacuum, you are competing against the other job applicants.

Create a stunning portfolio.  Do other projects.  Assuming you compare favorably against other applicants and the employer feels you can do the job, then you get the job.

Although Tom brings up another good point:  The fact that it was two attempts indicates the health condition is likely chronic rather than acute.  A one-time health condition is addressed once and you can move on.  A multi-instance or long-term health condition that impeded you multiple times means it is either something that needs to be controlled first or something that must otherwise be overcome in order to succeed in the workplace.