I'm 23 and a year into my PhD at Oxford and I've spent the past year consistently taking time off because I'm unhappy. I didn't really sit and think about my career options when I applied, I panicked and applied for three PhDs, got two of them but chose Oxford simply because it was closer to where I lived before (and for the name admittedly). I haven't made many friends here either and I'm very anti the Oxonian lifestyle.

I'm good at the work and its in a field I've been working in for a while, in fact I've received a lot of comments on the quality of my work and my supervisor said he's really impressed. I'm not that interested in the topic but the work would look absolutely fantastic on my CV and push me towards an engineering career. I keep thinking I could probably motivate myself through three years of indifference and boredom for the title... (Although this might be inherent in my personality).

Now there is an additional factor that makes this more personal and complicates things. About 4/5 months into my PhD, I started dating my best mate from London. I'm the happiest I've ever been but long distance is tough and he's working god awful hours on the opposite side of the city to his home. We toyed with me moving to London and commuting into my PhD but that's completely unfeasible. I wasn't that happy here before I started dating him and now I'm considering giving it up to go live with him and actually research a proper career I'm passionate about (and take some time out for my own sanity!)

I kind of feel like giving up a career for a relationship is embarrassing, especially this young when its just all for fun, but realistically I'd be giving it up for my own happiness. If I didn't have such a massive accomplishment ego, this decision would be a no brainer.

I think you already know the answer to your question.
For what it is worth, you dont need a PhD to work in engineering.
I am also not really sure why it is embarassing to ditch a job you clearly hate for any reason including for a relationship.
You already know what to do by the sounds of it.
No way should anyone waste 3 to 4 years just to get some letters after your name.
You can acomplish a lot more doing what you enjoy.

Maybe see if you can take extended leave from your PhD before quitting? What if you and the boyf spilt up next week or you can't stand living together?