Some really good advice has already been given and I'd like to stress Adam-M's point regarding a degree compared to certification. If you want to do a post-graduate degree, just do it, but if your (sole) goal is to increase your chances at landing a job, it won't have the desired effect.
The thing I look for in candidates is a combination of experience and potential. For experience prior jobs are best, but without those, hobby projects go a long way. So whenever you learn/try-out a new language or technique, try and make a (small) project out of it and throw it up on Github or something. Having an active project that's being used (and maintained) is even better, but that isn't trivially accomplished.
As for the "potential" part, I want to see a broad set of skills so I'm confident that the candidate can adapt. Say a candidate has some experience in Python, done a lot of C# (backend) and has done some web development (e.g. a SPA). I would feel confident tasking that person with writing a "(cloud) function" in Node.js (not as full time occupation, but as a one-off task). This wouldn't hold true if the person only ever developed in one language.
Now the above examples all focus on programming languages, but there are a lot more skills that are increasingly important to have in the field. Things like being familiar with Cloud providers (AWS, Azure, GCP), knowing about testing (unit testing, integration testing), DevOps related (CI/CD), Agile, SCRUM, etc... These skills are harder to demonstrate with "hobby projects", but you could blog about them or go the certification route. Quite frankly, I do think it looks odd when people have a lot of certificates compared to work experience, but that might just be me, as I find most certificates to be a bit of a farce (generally you need to learn for the exam rather than learn the subject itself).
I'm not sure what kind of jobs you apply for, but try and go broader. When hired as full stack developer, it could very well be that you end up doing practically nothing else but frontend work (most cases you can steer the direction you want!). It also depends a bit on the size of the company, the larger it is the more it tends to have clearly defined roles. Personally I prefer smaller companies as there's often a bit more freedom. Others like big corporate companies, as they like working their way up in those environments, but that often comes down to politics as opposed to showing what you can do. I guess the point I was trying to make is, the job as outlined in a vacancy is generally not as fixed as you might think.
Also on your CV, not sure how the Games degree is listed, but it might give people the wrong idea. Over here, there's also a Games degree, which is for 80% or more just general Computer Science, but might not look like that to others. Obviously never lie on your CV, but it wouldn't hurt if there are alternative names for your degree that focus more on the CS nature of it (or alternatively highlight parts of the degree that are general). On the subject of CVs, when listing anything I would focus on the skills or techniques you used/gained. You'd be surprised how often people just describe the product/project itself, instead of what they did on it and how.
(All the above is obviously based on my opinion and experiences, but might also be based on cultural differences as I don't live or work in the UK)
Lastly, stay positive and confident as that will get noticed.
The film cut a lot of the side characters "screen time". 
