Generally, if a country actually cares about preventing child labor, they prevent child labor regardless of the country of origin of the child, at least officially.
In this case, it seems more like a "we have no choice" scenario.
Maybe I'm optimistic. Although, the "kids" are older than 12, right? In that case, some cultures treat them as adults (like Dwarf Fortress, I should mention).
Yeah... it's kind of a "at least officially" thing. The archangel itself, as well as the gundams were... basically black ops. They were EA prototype weapon platforms being researched and constructed on a neutral colony (Think switzerland researching and building nukes for germany back in WWII as a hypothetical parallel, basically.), and the Earth Alliance already had a history of genocidal actions* (Dunno if it's come up in game yet or not, though -- checking over the text it looks like not. Still, it's backstory more than direct plot related.).
Things even up on the asshole scale later into the series, but right now... EA's basically genocidal racist assholes that wrecked some civvies and then got their teeth kicked in by superior ZAFT technology. Murrue and Mu's actually fairly decent individuals, but they're also basically officers in a close equivalent to the Nazi military -- though again, things even up later in the series on the asshole scale.
Which basically boils down to the
EA being jerkwads and probably not caring about child soldiers (and wait until later in the series, ahaha
-- also, the EA officers technically have/will have a mandate to murder Kira. Good thing for him he's useful
), but some of the Archangel's command staff probably do. Right now it's a last resort kind of thing, but in SRW plot especially it doesn't really pan out. They could have someone drop the kids off somewhere pretty easily.
As for actual legality... I've not actually seen enough of SEED to know if it came up in the show, but gundam being gundam probably not. Still... EA probably does have child soldier laws, going by support staff and enemy combatants and suchlike, and likely something closer to US law than anything else. ZAFT probably has a lower age limit, though. Unlike the AA crew that got shanghai'd, Athrun and crew are trained military elite and in roughly the same age group. There's also some other pressures on ZAFT for younger military (lower population, slower pop growth, etc. Plus the whole gene engineering thing. They go for more quality over quantity.).
And... I've now officially wrote too much, methinks. Still, general ramblings out, whee.
*
They nuked a civilian agricultural spinner -- Athrun's mom died in the attack.
)