The employer is required to pay gratuity within 30 days from the last working day.
If he does not do so, you can file a complaint with the authority under Payment of Gratuity act, which mostly is the labour commissioner. He will then ensure the payment is made.
There is a specific format for filing the complaint which is given in the act (but the commissioner office will also have the format)
You are required to file form I with your employer to claim gratuity. However if it is stated in your final settlement letter then commissioner will not insist on filing form I
You may wait for the 45 days that your HR team promised or proceed with the complaint, depending on how much you trust your former employer