The following is my understanding based on having worked in the UK.
Allowances don't roll over between tax years any allowance you didn't use-up last tax year is lost.
Within a year your allowance is split up between your pay periods and within a tax year unused allowances do carry forward. So what should happen is that you get both April's share of the allowance and May's share of the alloance applied to your May pay packet. After that you will be taxed as normal so assuming you stay in work to the year end you should have paid the correct ammount of tax. If you start working late in the tax year you may find you have several months of no income tax as you use up the carried-forward allowance.
Afaict whether that is what actually does happen depends on the competance of the people operating PAYE at your employer and whether you filled out the "starter checklist" (Formerly P46) correctly. At the end of the year its probablly a good idea to check the total tax paid is correct.