The PSA doesn't just apply to savings accounts. It relates to any interest earned, so the rules extend to other products, including interest-bearing current accounts. Interest earned on cash ISAs doesn't count towards your PSA, because it's already tax-free.
The government also introduced new rules to support the PSA, so banks and building societies are no longer deducting income tax from interest earned on savings and current accounts.
Regardless of when you opened your account, if the interest is paid after 6 April, you will receive it without tax deducted. This also applies when closing an account, where interest is added at the point of closure.