Firstly, I'm sorry for your loss. In order to transfer the ownership of the flat to your name or to you and your wife's name, you will need to follow the legal process as per the laws applicable in Hyderabad.
Since there was no nomination system in place, and the society has not issued any nomination forms, you will need to obtain a Succession Certificate from the court. This certificate will establish your legal right to the property as the son of the deceased father.
Regarding the letter given to the builder for nomination, it is possible that the society may accept it as a valid nomination document. However, it is best to confirm this with the society or seek legal advice.
As for the Will, even though it is not registered, it can still be used as evidence of your father's intention to leave the property to you and your wife. However, it is advisable to get it registered at the earliest.
To initiate the transfer process, you will need to submit the required documents, such as the Succession Certificate and the Will, to the society or the Registrar office. The society may initiate the transfer process, or you may need to go to the Registrar office and complete the formalities there.
Once the ownership is transferred to your name or to both you and your wife's name, it is important to ensure that all the relevant documents, such as the Sale Deed, Khata Certificate, and Property Tax receipts, are updated accordingly. This will help avoid any legal issues in the future, especially when you decide to sell the property.
In conclusion, it is advisable to seek legal advice to ensure a smooth and fool-proof transfer process of the property ownership to your name or to you and your wife's name.