Forgot to connect back to ground pin in this rough diagram
The examiner will only be turning my device on for literally 2 or 3 minutes (if even). Is it viable to use an arrangement of AA or AAA batteries to power these servos externally for this short period of time or will I have to use a wall power supply (provided I supply the Uno separately)?
It has been very common for years to power servos in model airplanes with a pack of four AA rechargeable batteries (which also power the receiver for the radio control). A few minutes of use on lightly loaded servos should be no problem.
Something that may be more problematic is feeding the power through a solderless breadboard - the contacts in the breadboard are not intended for high current.
4xAA alkaline batteries can supply 2 Amperes, and will power 1 or 2 servos if fresh. It is extremely unlikely that they will power six. NiMH(*) AA batteries might be able to do so, but don't count on it.
A breadboard will not handle that much current either, as the tracks will burn.
For six servos working at once, the power supply must be able to supply more than 6 times the servo stall current, as each servo draws the stall current every time it starts moving. It does not matter that it might move "just a few degrees".
If you don't follow this rule, you will end up with an uncontrollable mess of twitching and jerking servos.
What is the stall current for your servos? If you can't find it, post the servo model number, or a link to where you bought it.
6 x Hitec HS311s should be o.k. with 4 x AA NiMH rechargeables e.g. Eneloop or similar provided they are not very heavily loaded. And as NiMH are only 1.2V nominal so 4.8V for 4 they will also run the Uno via its 5V pin (NOTE: 5V pin not Vin or the barrel connector)
Alkaline batteries are nowhere near as good, not even Duracells despite what the bunny might say.