Your query involves three distinct aspects: (i) validity of a document notarized at the U.S. Embassy in India, (ii) whether apostille is required, and (iii) what banks practically insist upon in such transactions. I will address each clearly.

1. Whether a PoA / Affidavit notarized at the U.S. Embassy in Delhi is valid in India

Yes, such a document is legally valid in India, provided it is executed before a competent consular officer.

Under Indian law:

• Documents authenticated by diplomatic or consular officers are recognised under Section 3 of the Diplomatic and Consular Officers (Oaths and Fees) Act, 1948
• Courts in India generally accept such documents without requiring further proof of execution

If the son signs the PoA/Affidavit:

• At the U.S. Embassy/Consulate in Delhi
• In the presence of a consular/notarial officer
• With proper identification

then the execution is treated as valid.

1. Is apostille required in this case?

No, apostille is not required in your situation.

Apostille under the Hague Apostille Convention is required only when:

• A document is executed in one country and
• Needs to be used in another country

Here:

• The document is being executed in India (Delhi)
• It is to be used in India (Lucknow)

Therefore:

• No apostille from Ministry of External Affairs is required

1. Important practical requirement: adjudication / stamping

This is where most people make mistakes.

Even though the document is validly executed:

• If it is a Power of Attorney relating to immovable property, it must be:

• 
  Stamped / adjudicated in India under the Indian Stamp Act, 1899
  
• Typically within 3 months (varies by state practice)

In Uttar Pradesh:

• The PoA should be presented before the Collector/Stamp Authority for adjudication
• Proper stamp duty must be paid depending on the nature of authority given

If it is only an affidavit / no-objection declaration (and not a PoA):

• Stamping is minimal or not required
• But notarisation/authentication must still be clear

1. Bank’s concern – why they are asking this

Legally speaking:

• If the Will of the deceased brother is valid
• And the widow inherited and executed a registered gift deed

then the sons do not automatically have a right (assuming no challenge to Will).

However, banks take a risk-averse approach and often insist on:

• NOC / affidavit from legal heirs
• To avoid future title disputes

So this requirement is practical, not strictly legal.

1. What you should ideally do (recommended approach)

To avoid any objection from the bank or future purchaser:

• Prefer a No Objection Affidavit rather than a broad PoA
• Ensure it clearly states:

• Acknowledgment of father’s brother’s Will
• Acceptance of mother’s ownership
• No objection to gift deed
• No future claim over the property

• Get it:

• Signed at U.S. Embassy/Consulate in Delhi
• With passport details annexed

• After that:

• Notarisation is sufficient for affidavit
• Adjudication only if it is a PoA

1. Risk if not properly done

If the document is informal or improperly executed:

• Bank may refuse loan disbursement
• Buyer may delay or renegotiate
• Future title objections may arise

Conclusion

• A PoA/Affidavit executed before the U.S. Embassy in Delhi is legally valid in India
• No apostille is required since execution is within India
• If it is a PoA, it must be stamped/adjudicated in India
• For your purpose, a properly drafted NOC affidavit is usually sufficient and safer than a broad PoA

There are two different subjects that you have put together here in your question.

First one is an affidavit executed through US embassy in India. This affidavit is not a proper legal document to relinquish their rights in the property neither a NOC is a valid document to ratify the gift deed executed by their mother in favour of your father.

Second subject is the power of attorney deed. It is not understood that what is the reason or purpose to authorise a power agent through this deed. No doubt notarised power of attorney deed duly attested by Indian embassy is valid, however the attestation by US embassy in India may not be a proper option, they can very well get the power of attorney deed registered in India if they are in India for any reason. But it is still not understood that for what purpose and to whom they are going to give this POA deed, if they are authorising an Indian relative to execute the ratification deed through this POA deed, then there's no necessity for any affidavit.

The affidavit is not a proper document in this situation.

Last but not least, since the property was bequeathed by the brother of your father in favour of his wife based on which she has already transferred this property to your father by a registered gift deed, there's no necessity for a NOC or a ratification deed from the children of the deceased testator, the bank or the buyer cannot demand NOC or an affidavit and there's no necessity for any Power of attorney deed as well.

However in order to move on, children of the deceased can execute a NOC duly notarised and attested by the Indian embassy at US or the US embassy in India, that should suffice and solve the issue.