Dash Oil and Brake Warning light - how are they triggered in the start function?

Very simple. The key switch only has 3 outputs. ACC is powered in the "ACC" and the "run" positions of the key That line powers stuff like radio, wipers, heater

The "run" position of the key powers things like the gauges, the warning lamps, the ignition system, and the VR and alternator field, and in newer models, smog devices under the hood. THAT "run" power goes DEAD when the key is twisted to start. That means the warning lights go dead during engine cranking

In the start position, there are two separate isolated key switch contacts/ circuits live. 1 is the "start" power which involves the start relay and neutral switch, and energizes the starter solenoid. 2. is the coil "bypass" circuit which feeds full battery power to the coil side of the ballast resistor. If you have electronic ignition instead of breaker points, that power during cranking also BACKFEEDS through the coil ballast to provide power to the ECU for spark.
I only know my '72, '75, and my '90's pickups, but I have to disagree. The oil light on my cars is on during cranking, only goes out when the key is off or engine has pressure. I know this because if I crank long enough, it'll start winking out.

I'm not sure about accessory, I don't use that function (no cute girls will make out with me in my cars, so no need to sit with radio on and dash lights to grope by...)

My brake light ONLY comes on if there's a pressure problem or the parking brake is set. Low fluid doesn't trigger a light unless it's my later pickups. To be fair, I've never had low fluid in the cars, so I might be wrong on that. My brake warning light does not have a test indication. Only the oil light comes on at startup, if I've released the parking brake.