In certain regions and customs, marriage between cousins is allowed in India. According to the Hindu Marriage Act of 1955, marriage between cousins is considered illegal for Hindus. However, the exception is given if regional customs allow it. For example, in Southern India and Maharashtra, marriage between cousins is allowed because their customs allow it. In the rest of India, it is seen as incest.
Since you are from Tamilnadu where cousin marriages are allowed, I don't think there will be a problem if you marry your cousin. However, the marriages between cross cousins are allowed. If she was your father's sister's daughter, you might not face any difficulty. Since she is your father's brother's daughter, I don't feel it is legal. You need to confirm it from a lawyer before you proceed with your proposal. Or talk to your elders to know whether such marriages are allowed in your custom. If your custom permits it, it is legal too.