Every employee should receive a P60 every year giving details of their total pay and tax deducted for the income tax year ended 5 April this month. The deadline for issuing P60s is the 31 May, but it would make for good relations with employees to issue them much earlier than that. The earliest date would be 6 April, but employers are expected to have done all their returns to the Revenue by 19 April, which is today. Once your payroll is finalised, there is no reason why you should not issue the P60s at once. Employees may need them to complete a tax return, for a student loan application for a child, for a mortgage application or for some other purpose which is important to them.
Sensible employees will keep both the P60 and their last payslip before 5 April. Payslips where we act as as the payroll bureau have running totals to check to the P60, and also summaries of auto-enrollment pension contributions. If an employee wants to track down a pension pot, then the information is all on our pay slip, but not necessarily on the P60.