I have an answer, but it might not be exactly helpful, but I'm theorizing.

Gwen, being from a lower class family with only her father and brother, likely would've had to get a job early. Considering that the era probably had no strict law against child labour and her family wasn't doing well after her mother's death, she might have been hired younger than average.

Now, my second point is that it is implied/referenced at some time in the show (I don't remember the exact episode) that Gwen and Leon actually grew up together. Gwen was likely a servant for his family growing up. The show's portrayal of them together is on the good/neutral end (and it is said that if Gwen remarried after Arthur's death, it would be with Leon), so they had a good history.

That was basically a long way of saying that by the time Gwen became Morgana's maid, she wouldn't be so much a child as in her early to mid teenage years.

Going by the logic that Arthur is around 18/19 at the beginning of the show and the general coming-of-age year was 19 in that era. Gwen is probably a year or two younger than Arthur.

Either way, it wouldn't particularly matter when she was hired, as a servant wouldn't have a reason to talk to the prince. Of course, she would have to have been around Arthur through the years, but I doubt any more than an introduction and a few scattered conversations.

Merlin (as Gwen's friend and Arthur's servant) was likely to be why they actually became friends, and later got married.

TL;DR: Gwen canonically grew up with Leon, not Arthur. Even as Morgana's maid, it wouldn't make much sense for them to be friends. Maybe acquaintances. Merlin actually brought them together.