This question is a regular subject for calls, especially round the latter part of December each year.
The simple answer is pretty straight forward. The complex answer, funnily enough, is not. However, if you have to ask this question then there are a few other basic questions that you need to ask yourself for the bigger picture.
If you have worked for your employer for over 12 continuous months and you have a positive leave leave balance then you are entitled to any unused annual leave. Legally, you are not entitled to any annual leave until the first anniversary of your continuous employment. However, many good employers will accrue annual leave for you from the first day of you working for them and they will effectively let you use this leave even though legally they don't have to.
The issue this raises is that why would an employer not let you use annual leave you have accrued but you have not worked for a period of 12 months for them? Generally having employees with holidays owing is a nuisance for employers. Firstly they have to set money aside for this that technically is not theirs and secondly, they then run the risk of you wanting holidays when it does not suit them, thus there is the possibility for workplace conflict. Thirdly, traditionally, employees are at their least productive over the Christmas new years weeks. What I have to say next is a mass generalisation and should not be read as gospel, as some large employers have blanket rules for various reasons so my next statement may not apply. The most common rule we find employers not giving employees leave over Christmas is due to the employer not valuing the employee and possibly the employer wanting to get rid of that employee. You will know in your heart if this statement rings true for you or not. If this statement has hit a nerve then it's time to start seeking new employment so that you can leave on your own terms, not your employer's terms.
Should you want to know in more details about your entitlements for working public holidays over the Christmas break then best to call the Ministry of Business, Innovation & Employment (formally the Department of Labour) on 0800 20 90 20. Usually the time people want to know about this is when I am being festive myself and the chance of you getting a straight coherent answer out of me is quite remote.