I believe the system design would be very similar to standard aircraft ejection systems.
The main oxygen supply is airframe mounted and connected to the aircrew with a hose from the structure to the seat via a quick disconnect that is separated via a lanyard that pulls the airframe hose clear as the seat departs. The oxygen line is sealed to atmosphere.
The pilot remains strapped to the seat via the chute harness and breathes the emergency system oxygen mounted on the seat. There is a barostat that that keeps the pilot attached to the seat until the altitude is low enough for the pilot to breath. This barostat also keeps the main chute stowed and the drogue chute is deployed to stabilize the seat during decent.
Once this altitude is reached the barostat allows the main chute to deploy, the harness disconnect from seat and the seat falls away as the main chute deploys (at the lower, safer altitude). At this time the hose from seat to harness is pulled free via a lanyard and the pilot breaths a combination of oxygen and air (there would be a depressurization however the pilot is at a safe altitude).
The seat base holds a package containing a life raft that is pulled clear as the seat falls away. This is connected to the harness and the pilot can disconnect during decent, as this package falls it is connected via a lanyard that fires the life raft to inflate and remain connected to pilot when they splash down.
There are many variations to this sequence and barostat altitude settings and time delay vary. The Gemini familiarization manual is available online and gives a description of sequence and function.