Strictly speaking, the design industry (assuming you meant fashion design, graphics, product design, etc.) and software development don't require you to have a degree of any sort. Most allow you to get into the industry with minimal qualifications, so long you have a track record or be willing to work from the ground up.
Don't get me wrong, a MEng in Mechanical Engineering will get you very very far, but it's not necessary should this not be something you really want. Mechanical Engineering can be applied to product design engineering (as opposed to just drawing nice pictures of products) and can be used to get into software engineering (as programming should be included in the degree). It's also recognised worldwide if the degree is accredited by iMechE (see:
https://www.imeche.org/,
https://www.imeche.org/membership-registration/become-a-member/academic-requirements,
https://www.engc.org.uk/education-skills/course-search/accredited-course-search/) i.e. it should be.
I would agree that you should redo your GCSE English and Maths, since they would hold you back a lot (and you seem significantly more capable than the grades they seem to suggest). Personally, I would go for a degree apprenticeship in either design or software engineering after your GCSEs, but that's your decision to make.
Most vocational/adult colleges in London and Cardiff should offer you the opportunity to retake your GCSEs, and most of them should allow you to do this for free. I am not sure why you had difficulty finding them. For example:
https://cavc.ac.uk/en/subject/gcses#GCSEshttps://cardifftutorcompany.co.uk/2023/03/16/free-gcse-maths-english-courses-adults/https://www.adultlearning.wales/en/about/our-courses/flexible-maths-and-englishhttps://www.southbankcolleges.ac.uk/courses/english-maths-for-adultshttps://www.cnwl.ac.uk/student-support/free-courses-for-adultshttps://www.waes.ac.uk/department/english-and-mathshttps://www.wlc.ac.uk/search-results/?keyword=gcsehttps://www.capitalccg.ac.uk/courses/english-and-maths/635/There should be plenty more in London, but I the above should be more than get you started (I google the terms "GCSE English maths adult college [location]").
If you want to get into design, then more cases than not, you should be in London for this. In terms of software engineering, it's kind of spotted around the country (to my knowledge at least).
What I don't particularly recommend is to go back to college to do a course in software engineering or design, as they would very unlikely to help your case. If you feel particularly inept at programming, I suppose a Level 3 course in computing (with a lot of programming content) can help to an extent, but I would then focus on networking, attending career fairs/presentations, etc. to get the role in software development.
For design, I would focus on putting together a portfolio should you need it for entry level roles.
Should for any reason you wish to change your degree and go into either computer science or design, then it's your choice to make. However, I don't think they would particularly help you get the roles that you want.