Melanie Shee's reply:
Unfortunately many employers do tend to employ graphic designers who have a degree in a relevant subject area which may account for why you are having a problem finding an apprenticeship as there are not as many available in this sector than others such as Digital Media and Marketing.
However, you do not necessarily have to undertake a degree course as there other courses available such as that Shillington College who offers a three-month (full time) or nine-month (part time) portfolio-based graphic design programme. In fact, a former colleague of mine has just completed this course and is currently working in a design agency in Northumberland. However, although he is now competent in some of the key packages such as Adobe InDesign, Adobe Illustrator, Adobe Photoshop, Adobe Acrobat and Sketch he feels he has missed out developing specific creative skills and lacks the `creative’ aspect his role requires. He is now considering going to university to develop these skills via enrolling on an arts foundation course so please bare this is mind. It also a highly competitive area and you will be competing with graduates who have developed their design portfolio.
If you decide to undertake undergraduate study you could study for a specific graphic design course or it could be in any design-related discipline, such as, photography, illustration or visual art, but a specific graphic design degree will be probably be the most useful as you will gain the specific skills and knowledge for the role as a graphic designer. There are a wide range of degree and foundation degree course available so if you do decide to go choose wisely. I would suggest you look at the Which University http://university.which.co.uk/ for free and impartial guide to courses.
If I were you I would look at current job descriptions for a graphic designer to see what employers are looking for if you wish to be employed by a company as opposed to working for yourself. I would also consider contacting a few local design companies and ask them if you can pop in for a quick chat to see to see what skills and personal attributes they are looking if they were going to employ someone as this may help you make a decision of what route to go down.
If you do your research and decide University is not for you could take a look here https://www.notgoingtouni.co.uk/ as they do advertise apprenticeships.
I hope this is of some. Best wishes and good luck.