Suppose there are 2 banks, A & B. Bank A has merged into bank B, hence, Bank B is the anchor bank & bank A does not exist anymore.
How is a salaried person who has submitted the Swift Code & IBAN number of Bank A to their employer affected?
It doesn’t matter whether the banks are private or public. The acquiring bank will work out its own policy regarding account numbers and whether it would continue maintaining the acquired bank as a separate entity.
Either way, you should ask the bank to provide you with the new details you need to give to your employer. Even if they have decided to retain the old numbers, it’s good to know that for certain.
It's possible that your IBAN number will change.
When bank A continues to operate under its old brand even though it is now 100% owned by bank B, then they will likely retain their own bank code and thus all customers will keep their IBAN numbers. But when bank A no longer exists as a separate entity, then it is indeed likely that they are going to unify their bank account numbers under the bank code of bank B. So all customers of bank A will receive a new IBAN number at bank B. A cursory internet search finds several articles from banks where they inform their customers about such a change.
Bank B will likely keep processing payments to the obsolete account numbers and redirect them to the new number for a while, but they will likely not do so for all eternity. Banks do have ways to inform people who regularly send payments to their accounts that these accounts have changed. I don't know how well the accounting department at your employer works, but personally I would not rely on them to react to these messages on their own. So you should inform your employer (as well as everyone else who regularly sends you money or has a mandate for direct debit) that your bank account number has changed. Any SEPA mandates you signed will become invalid with the account change and will have to be renewed.
Your bank should send you detailed information about your new IBAN number, when it will become valid and when your old number will become invalid.
SWIFT codes (aka BIC) are only relevant for payments where the receiver is in a country outside of the SEPA zone (all of the EU and a couple more countries). For any payments within or between SEPA countries, it's redundant because the bank is also encoded within the IBAN.
Everything might happen.
In short, every change you will experience will be communicated to you ahead of time.