Could I afford to move out given my current finances?

Surely this depends on whether you get a job and what the salary is... a lot of grad schemes provide relocation assistance so some of that initial expenses of moving is removed.
How long of university do you have left?

Would you be able to get a part time job while at university to save up? E.g. supermarket, retail, tutoring, restaurants etc
Also, I would recommend working out how much living in your own/with a roommate would cost you.
Then see how much your job would pay and adjust where you live to suit.
Be aware that certain places are more expensive than others. Obviously London is very expansive, but so is Bristol and a few other places.
You need to plan and see what you can afford. Remember you will have like a month from starting work to being paid (I think), so making sure you can manage that and you need to create an emergency fund for any unexpected spending,

2 months, I finish in May

So have you got a grad job lined up then for September?

I have a job right now, but only part-time. I'm on roughly 13 grand a year.

That's not enough to live on, sorry. Maybe if it were 1990 (even then it's unlikely... 1970 perhaps) but in 2025 no chance.

I know, hence why I'm trying to save up. I'm basically just dumping a lot of my pay into my savings whilst also looking for a higher-paying job elsewhere.

So right now I'm thinking about moving out of my hometown after I've finished university. My plan is simply to move to wherever I get a job, but I do have one slight issue: I really do not have much money behind me whatsoever. I've never been employed, and have been living off my maintenance loan at uni for the entire time I've been here (in other words, I've been living off 7-8 grand a year since 2022). Although I've managed to make it work, and I've never struggled with rent or anything, I'm aware that it's a whole different ball game post-university, with the costs of all the bills and stuff like that. Moving out is the best decision for me career-wise, but I don't want to end up in a position where I get a good ass job, move out, then suddenly realize I can't afford it.

Totally get where you're coming from, but honestly, surviving on that maintenance loan proves you’re already better at budgeting than you think. When you start interviewing, don't be afraid to ask if the company offers a relocation package or a signing bonus to help cover those heavy upfront costs. You should definitely look into flat-sharing for the first year since bills and council tax can be a nightmare when you live alone. It might be worth picking up a quick side gig right now just to build a tiny safety net before the big move. Just make sure you actually have the job offer signed and sealed before you commit to a lease!