Is it obligatory for the questioner to fast on his behalf and on behalf of his deceased wife to complete the expiation for Hajj which was not fasted ten years ago?

Your fasting for three days during Hajj due to the unavailability of a sacrificial animal is a valid act. However, your delay in fasting the remaining seven days is not appropriate, and you should hasten to make up what you owe. What is now obligatory upon you is to fast the seven days on your own behalf. As for the woman, if none of her relatives fast on her behalf, then you must feed a needy person for each day, based on the Prophet's (peace and blessings be upon him) saying: "Whoever dies while owing fasts, his guardian should fast on his behalf."