It might well affect their perception of their parents' relationship though. ("Oh, btw, our marriage wasn't really a marriage.") This isn't how real relationships work.
That problem could be cured by offering a correct explanation to the children. The explanation would include a comparison of secular/civil law marriage vs. sacramental marriage. The two are different.
Annulment is a quasi-legal nicety designed to side-step the "problem" of a marriage breaking down. If the CC accepted that divorce happens, and that there are really far worse things going on in the world, they wouldn't have to cobble together the nonsense that is annulment.
The Catholic Church does accept that divorce happens. It correctly understands that marriage and divorce are secular/legal/civil law matter, but holds that marriage in the RC church is a sacrament as well as a secular/civil law matter.
The CoE and other protestant denominations don't hold a view of marriage as a sacrament. The RC annulment process is separate from the secular/civil law divorce process and seeks to answer questions related to the sacramental nature of the marriage. These questions are entirely separate from the question of whether a marriage was legal in the civil law sense.
To be a valid sacrament of matrimony in the RC church, both parties have to meet certain conditions as they decide to marry and prepare for marriage. The RC church looks at the circumstances that existed in the lives, minds, and hearts of each of the parties at the time the wedding took place as it comes to its decision on annulment, which is a statement that the marriage was sacramentally invalid (canon law), not legally (civil law) null and void.
It's not a nicety designed to replace civil divorce or to deny that the civil law exists, that civil marriage exists, or that divorce exists. The RC church sees the civil law as a fundamental and indispensable element of society. The annulment process does not go ahead until a civil divorce has been granted. You have to present proof of secular/ civil law divorce along with your other paperwork when you file or respond to a petition for RC annulment of your sacramental marriage.
There are two jurisdictions at play in the case of civil marriage and sacramental marriage. One is the civil law, the law of the land. The other is Canon Law, the law of the RC church. They are completely separate systems, designed with completely different ends, and with completely different terms of reference.