These two features would stack, allowing +2 to AC
The Blessings of the Forge feature requires nonmagical armor to grant a +1 bonus to AC, and although the Warforged racial feature grants +1 to AC, it isn't armor, nor is it magical. Because the source of this +1 bonus to AC isn't magical armor, there's no reason it wouldn't stack with magical armor whether created via Blessings of the Forge or not.
A good comparison would be the Defense fighting style (available to fighters, paladins and rangers, PHB p. 72, p. 84 and p. 91 respectively):
While you are wearing armor, you gain a +1 bonus to AC.
This, again, isn't to do with magical armor, it is simply something that you benefit from while wearing armor, even nonmagical armor, so would stack with magical armor, even armor that is only temporarily magical thanks to Blessings of the Forge.
As a counterexample, consider Artificer Infusions
The Artificer class, from Eberron: Rising from the Last War (p. 57), has the class feature Infuse Item, which describes how you apply the class feature to items under "Infusing an Item":
Whenever you finish a long rest, you can touch a nonmagical object and imbue it with one of your artificer infusions, turning it into a magic item.
If we consider the Artificer Infusion called Enhanced Defense (p. 62), which says:
Item: A suit of armor or a shield
A creature gains a +1 bonus to Armor Class while wearing (armor) or wielding (shield) the infused item. This bonus increases to +2 when you reach 10th level in this class.
So, using the Enhanced Defense infusion on a nonmagical piece of armor turns it into magical armor, and using Blessings of the Forge on nonmagical armor also turns it into magical armor (it says "the object becomes a magic item" in the quote in the question), so these two features cannot be used on the same armor, since as soon as one feature takes effect, the armor becomes magical, and therefore is an invalid target for the other feature, each of which requires nonmagical armor only.
(NB: I'm intentionally restricting myself to a suit of armor here, ignoring shields for the purposes of making this point.)
So, because the Warforged racial ability makes no mention of magical armor (or armor at all, not in that quoted bullet point, at least), there is no reason why this bonus would stack from the bonus granted by Blessings of the Forge.