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A suit of Studded Leather sets AC to 12 + Dex. A suit of magical +1 Studded Leather gives a +1 bonus to AC while you wear it, for a total of 12 + Dex +1. Mage Armour gives an AC of 13 + Dex. This is straightforward enough, but if I put on +1 Studded Leather and then cast Mage Armour, is my AC 13 + Dex + 1?

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First line of Mage Armour;

You touch a willing creature who isn’t wearing armor

Last line of Mage Armour;

The spell ends if the target dons armour or if you dismiss the spell as an action.

So, you can't even use Mage Armour on yourself if you're wearing or don armour, be it magic studded leather or not, let alone gains it's benefits in your scenario.

If we forgo this limitation and pretend it is possible, we can look at page 14 of the PHB;

Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.

Which allows us to choose between the Mage Armour's base AC or the suit of armour's base AC. See this question for a better explanation.

Then there's the +1 bonus from the magic armour itself.
Does the armour increase the set value that the suit provides or is it in addition to?
The previously linked question provides some insight into that as well, but let's also look at the description of the magic armour (DMG pg.152).

You have a bonus to AC while wearing this armour.

There's also this question which explains that Mage Armour and Shield stack (Shield having the same descriptor [you have a +5 bonus to AC]) so there's no reason to think the same rules couldn't be applied here.

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