When you joined in RRB and after a month, you got a response from the Indian Bank regarding your document verification hence it implies that you have not made any response while remaining in the present job and as such NOC ( No Objection Certificate) would be required.
In case, you are asked to be present with all your related documents for the verification purpose during the process of selection, help the officials in this process by producing all such documents.
Once you are selected for the post of Indian Bank, you would be required to submit a resignation to RRB authorities indicating that you have been selected in the Indian Bank and as such your resignation should be accepted. You will be served a relieving letter provided you clear the dues if any to the authorities of the company. Moreover, you will be intimated regarding your probable date of your relieving. The same is to be communicated to your next employer.
Even if your date of joining for the next post exceeds the period when you were supposed to join, it would not matter much if you have made an advanced intimation to your next employer regarding your date of joining and such a date of joining has been accepted from their end.