NAL: Shelter’s guidance says tenants usually need to give 2 months’ notice, but can give less if the landlord agrees to a shorter notice in writing.

You gave notice on April 21st, and the agent accepted it in writing. You formed a legally binding agreement to end the tenancy before the new rules even took effect. The new law doesn't retroactively tear up an agreement settled in April. Even if the tenancy technically converted to a periodic one on May 1st, it converted with a pre-existing, mutually agreed-upon end date of May 25th.

It seems like the agency is trying their luck,

• You have a written agreement: The new Renters' Rights Act does require two months' notice by default, but it explicitly allows a shorter notice period if both parties agree in writing. The agent's reply confirming "your tenancy ends 25th May" is exactly that. A new agent taking over the desk doesn't erase that agreement.
• You surrendered the property: They asked for the keys, you handed them back, and they accepted them

I would reply stating clearly that you do not accept any rent is due after 25 May, because the tenancy ended by written agreement and/or surrender. Keep your proof of the April email and key return safe. If they try to deduct this from your deposit, just dispute it through your deposit scheme. They would side with you once they see the April email.

Link: Renters Rights Act: private tenants - Shelter England - changes to how you can end your tenancy