The employer is right in that the new tax year means your P45 won't be needed for tax purposes as codes went up on 6th april - you just say you haven't worked for anyone else and fill in a P46 (ticking section A) which puts you on 522L. This means you avoid emergency tax.
However, you have a right to have a statement summarising your earnings. if they wrote you off the system on the 31st March then a P45 will be generated and will list your earnings and tax payments
If they didn't do it until after the 6th April, a P60 will have been generated so ask them for that instead.
If you earn below the tax threshold and are happy that you haven't overpaid tax (your payslip should list total earnings and total tax paid to date) then its not a big deal. However, if you think you might have done, then you should get ypu P45 because you will need it to claim money back from the wonderful people at the revenue.