It...appears so, but probably shouldn't.
I suspect this is not intended and is yet-another artifact of things sometimes being worded unclearly in D&D 5E...but, well. Here we are.
The simple facts are that +1 Leather would put you at AC 12 + Dex, which is less than the AC 13 + Dex that your natural armor gives you, so by
You can use your natural armor to determine your AC if the armor you wear would leave you with a lower AC.
You may use your natural armor to calculate AC. It's a conditional override of the prior statement. "When you aren't wearing armor, calculate your AC this way--if your armor renders a lower AC, use your Natural even though you are wearing armor." Then, when we look at the description of Armor +1, it says
You have a +1 bonus to AC while wearing this armor.
For comparison, a Ring of Protection says...
You gain a +1 bonus to AC and saving throws while wearing this ring.
And a Cloak of Protection says...
You gain a +1 bonus to AC and saving throws while you wear this cloak.
It's the same exact verbiage. It has no bearing on how you calculate your AC--it's just "When you wear this, you get +1 to AC."
If it instead said something like "The AC value of this armor is increased by 1" then it would be clearer. But that is not what it says.
Again, I suspect this may not be intended...but with the rules as they are written--yep.
The basic problem
This is, naturally, a quirk of wording and not something I'd allow at my table.
In my opinion, the basic problem is how the +1 Armor is phrased.
The Lizardfolk's Natural Armor rule makes sense in how it works--it's not like their skin gets less durable if they put a sheet of leather over it.
But the phrasing of +1 Armor being identical to the phrasing of Ring/Cloak of Protection is causing this confusion.
From the same 'it's what makes sense" perspective of the Lizardfolk's armor--it makes sense that the Armor's +1 bonus applies to the armor itself, rather than somehow making its wearer's skin tougher.
Ultimately, this is mild cheese. You're exploiting a quirk in the exact phrasing to turn a suit of +1 Leather into the equivalent of +2 Studded. If your DM lets you get away with this, then sure...have fun.
But I feel like it's pretty clearly things not functioning quite as intended.