Of course you entitled to holiday!! As a p/t your holiday would be worked at a pro rata basis. Are you sure you haven't been entitled to it but just never taken it, as most companies if you don't take it by the end of the working year then you lose it.
You need to keep ringing hr, and your boss should be looking into it for you, but you may need to nag him/her.
How many holiday days did you take last year and were you just not paid for these days?
We don't pay sick pay at our company but you do go on to ssp after a certain amount of days (3 or 5 can't remember)
Your entitlement is worked out on how many days per week you work, I am part time but work 5 days a week so get 30 days per year inc bank holidays. Do you work on bank holidays? They would be given as lieu days if you work them.
But you have a head office? This must be a reasonably sized company, even if your place of work is small.
You do need to sort this urgently. If you're not getting holiday pay, are you sure they have you on the right tax code and are paying your NI contributions?
Holiday pay is a basic right of employment. You earn something like an hour's paid holiday for every 8 hours you work (or something similar!). Without paid holiday, you're basically earning less, as you're having to subsidise your own holidays.
Have you any written correspondence from work? Maybe about your salary or place of work? Rules of conduct? Anything like that could prior you have a de facto contract ie that you are an employee.