You'll probably turn into a Dragonborn immediately, but it depends which spell your DM considers 'more potent'.
- As long as the spells duration's overlap, you won't revert to your original form.
If you dismiss your first casting before casting the spell a second time, then your normal form would be visible. Both dismissing and casting Disguise Self requires an Action and you cannot use two Actions simultaneously. Even when a feature allows you to take two Actions in a combat round, the actions are still understood to occur consecutively, not concurrently.
If you dismiss the first Disguise before Casting a second Disguise Self spell then they aren't concurrent spells.
- Even if the two spell's durations overlap they still wouldn't overlap visually. Disguise Self doesn't simply produce a magical hologram.
While it might be amusing to think about the intangible hat you can create with disguise self, it isn't simply a magical hologram. If Disguise Self worked like a hologram then it would be impossible to disguise yourself as someone shorter or thinner than your original form - your head or your stomach would end up poking through the disguise.
'The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies[...].' (PHB p. 205)
So, while it might be funny to imagine two overlapping holograms, a shorter stockier dragonborn, with an elf's head sitting on top, delicate fingers semmingly extending from the dragonborn's finger tips, magic disguises don't work like holograms. While both spells are active, one would simply take precedence over the other.
- But which disguise would be the dominant one? Which spell would be the 'most potent'? In this case it's up to the DM
Let's refer back to that rule again...:
'the most potent effect—such as the highest bonus—from those castings applies while their durations overlap.' (PHB p. 205)
Diguise Self requires an Intelligence (Investigation) Check against spell save save DC, from a creature that wishes to work out whether or not you are disguised.
If two different casters had cast Disguise Self on the same target, it seems logical that the spell cast by the one with the higher spell save DC would be considered more potent and take precedence, until it was dismissed or its duration expired.
However, when the spell save DC is the same, either between two different casters, or because the same caster has cast Disguise Self twice, there is no guidance as to how 'the most potent' spell should be adjudicated.
A DM should choose a method, make sure the players understand it, and then ensure they apply it consistently.
I personally would rule that the most recent spell cast takes precedence, effectively overlaying the previous casting.
- Is there anywhere else I could look for guidance on this?
For evidence of a similar (but not definitive) ruling, that some might choose to consider authoratative, you might be interested in this tweet from Mike Mearls:
Q: Could you use multiple castings of Disguise Self to layer a disguise and force multiple checks against DC?
A: No, I'd only apply most recent casting.
See also, this tweet from Jeremy Crawford, which agrees with this general principle but isn't specific to Disguise Self.
- So, when does the Elf turn into a Dragonborn?
If the second spell is ruled by the DM as more potent, then the PC will change into a dragonborn as soon as the second spell is cast. This is what I would rule happens.
If the DM rules that the first spell is 'more potent', the PC would only appear to change from an Elf into a Dragonborn when the first spell's duration expires.