I presume then that those nearest just sling BA on over their working rig and crack on whilst another group suits up in more robust fire fighting gear?
The very first people at the scene will be expected to extinguish the fire.
They will be relieved within two minutes by a pair of ship's company (pre-nominated) wearing BA and a bit more protection.
Those two will be relived within a further six minutes and thirty seconds by five members of the ship's company (also pre-determined). If the five man team (which can include women, obvs) can't extinguish the fire, or have to close the compartment down, the ship will go to emergency stations.
At that point everyone goes to their emergency station: the first five people who arrive at the repair posts (who could be any specialisation or rank, including Officers) will be nominated as the immediate relief party, get dressed in fire fighting equipment. They will then immediately head to the scene of the incident (if the door to the compartment remains open) and fight the fire, or will remain ready at a muster point until all the SOPs are complete before re-entering the compartment.
The guys who are pre-nominated are part of the Standing Sea Emergency Party; as the XO (2 i/c) I was effectively i/c recce for the SSEP C2 process. The most junior guy in the SSEP can have been on-board for as few as three weeks.