What is the disadvantage of being illegitimate in the eyes of the Church? That is to say, what is the burden of illegitimacy today?
As far as I can see in the Canons and the Catechism, there is no disadvantage or burden. I can find nowhere in either where it says that the sacraments are to be denied the illegitimate. This might be expected: Canon Law strives to be compassionately just (or justly compassionate) and it's manifestly unjust and uncompassionate to deny a child the ministry of the Church because of an act which it could have absolutely no control over. A child is a gift, and a person from the moment of conception (CCC 2378): how can the Church deny a child of God who is blameless for their situation?
However, the notion of legitimacy hasn't entirely been abandoned, because Canon 1137 specifically defines it. The Canons do not define illegitimacy, nor do they specify what happens when marriages are invalid or annulled (except by implication in Canon 1137).
In the instant case, it appears that the marriage is at least a putative marriage — one that has all the signs of being the real thing. Canon 1137 unequivocally states that children born of a putative marriage are legitimate. Given that there appears to be no way of becoming illegitimate, presumably the annulment will have no such effect. (But your correspondent will need to ask the question of those dealing with the annulment, as they will have access to all the circumstances.)
This is similar to statute law, at least in England. Divorce does not make legitimate children illegitimate. There is a difference in statute law between an invalid marriage and an impeded marriage: a lawful marriage where there is invalid intent does produce a marriage, but can be ended. A marriage where there is a legal impediment is impossible, and is void ab initio; and its children are statutorily illegitimate. However, it appears that the Church would consider its children canonically legitimate.