Is a wife obligated to cover her face in front of her sister's son after he reaches the age of fourteen, knowing that she participated in his upbringing, and this might cause embarrassment?

If your sister's son reaches the age of fourteen, he is considered an adult or close to adulthood and a teenager. Therefore, your wife must observe hijab in his presence. Obedience to Allah takes precedence over any embarrassment, and there is no obedience to a creature in disobedience to the Creator.