Being unfairly denied overtime + being told I'm part-time despite working full-time hours for 15 months.

You need to ask them for your (written) "statement of employment particulars", which they should have given you within a few months of starting - if you don't have a written contract, you have to have one of these. This should then state your hours (39 hours would be considered full-time) and give you something a bit better to go off.

https://www.gov.uk/employment-contracts-and-conditions/written-statement-of-employment-particulars

My supervisor questioned it, and was told … I was "part-time staff".

You say you don't have a contract, so I'm not sure how you'd argue this in court, however zero-hours contracts are now predominant in retail work.

That means that neither the employer nor the employee has an obligation to give hours to the other - the hours are "flexible" and might, in theory, be negotiated on a week-to-week basis.

In practice, employers have more power to dictate terms than employees do, so employing all their staff on zero hours contracts better allows them to fuck you over.

If a shop has 3 employees working 40 hours a week then they are left short-staffed if an employee is ill - if they have 4 employees each working 30 hours a week, then they can ask the others to work a few extra hours to make up. Retail bosses don't like employees clocking more than 40 hours a week because overtime rates then kick in - better to have a bunch of zero-hours employees working a regular 30 hours a week, each glad to do a few extra hours.

Zero-hours contracts also allow bosses to punish employees arbitrarily - if you refuse to work overtime, call in sick, demand your agreed holiday or if he just takes offence at something you say, your boss can just reduce the number of hours he rotas you in for next week. Is it possible you inadvertently pissed someone off?

As you say you don't have a written contract, it may be that your implied contract makes you a full-time employee. I don't know much about this area of law - I think that's maybe a possibility, but I don't know. Good luck enforcing it, though. Your employer is going to say they took you on on a zero-hours contract, I bet.

You could speak to a solicitor who specialises in employment law - I've written previously, here and here, about finding and choosing a lawyer. I fear they may tell you that you're unikely to achieve any redress. Citizens' Advice and ACAS can also be good, I hear. I fear that your best resolution will be to get another job. You should write to your MP and tell him of your experiene, asking him to support stronger employee protection.

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https://www.gov.uk/part-time-worker-rights

Part-time workers are protected from being treated less favourably than equivalent full-time workers just because they’re part time.

There is an exception about overtime rates not being applied until you've met the usual full-time hours, but it sounds like you are meeting those hours already. There is no single definition of "full time", however, and it is probably harder to discuss whether you are full or part time without a contract. You definitely sound full-time to me.

But... yeah, this sounds like you're being "treated less favourably" to me. You should ask for their reasons in writing if you want to argue it (reminding them that must respond within 21 days, citing that link)