Airport emergency services are presumably trained in this, but since a plane cannot stop easily (or not at all on takeoff after V1), I seem to remember the general rule is that even emergency vehicles with lights and sirens on give way to planes, and don't enter runways without permission from the tower.

In the audio released by the BBC, the fire truck DID get permission from the tower to cross something, I can't tell if it was the runway in question. However, to cross the red runway lights if lit, you normally need that spelled out too something like "truck one, cross four delta, cross red lights". This did not happen on the BBC audio, which could mean one of many things.