Dear Jayesh,
The Partnership Firm should give a written letter to all its clients about the change in the status of the Organisation and also the new PAN of the partnership firm. The P'ship Firm should obtain an acknowledgment from its clients in regard to the receipt of the letter sent by your organisation.
As mentioned in your query, the TDS amount cannot be claimed by the P'ship firm, also the Form 16A bears the PAN of the AOP. Hence, from now onwards it should be seen that the deductor deducts TDS on the correct PAN, otherwise you won't get the TDS credit.
Regards,
Devendra P Kulkarni