Is my marriage to a Hindu woman, after my conversion to Islam, and then her conversion to Islam and the performance of a new marriage contract without observing the waiting period (iddah); considered permissible, and is my son from her legitimate?

It is not permissible to marry Hindus, as they are polytheists by consensus, based on the Almighty's saying: {And do not marry polytheist women until they believe} and {And do not hold onto the marriage bonds of disbelieving women}. What you did by re-contracting the marriage after her conversion to Islam is correct.

As for stipulating an ‘iddah (waiting period) from the previous invalid marriage: 1. If you believed the validity of your first marriage to her out of ignorance: Renewing the contract is sufficient, and no ‘iddah is required, because lineage is established in this case, and the wisdom behind the ‘iddah is the preservation of lineages. 2. If you knew the marriage was forbidden: The first marriage is invalid, and you must repent. There is a difference of opinion regarding the necessity of ‘iddah: Some scholars do not obligate ‘iddah, considering that unlawful fluid has no sanctity. Others obligate it (or istibra’ - waiting to ensure the womb is clear) to ascertain the purity of the womb and preserve lineages. The preponderant opinion is istibra’ with a single menstrual cycle only to ensure the womb is clear, as preferred by Shaykh al-Islam Ibn Taymiyyah and Ibn ‘Uthaymeen.

Accordingly: If you believed the first marriage was permissible: You did well by renewing the contract, and nothing else is required of you, nor an ‘iddah. If you believed it was forbidden: You must repent, then renew the contract after you separate from her and she has one menstrual cycle to ensure her womb is clear. If you are certain she is not pregnant, nothing is required of you. If you renewed the contract before istibra’, then renew it again now, and your child will be attributed to you.