The implication seems to be that air was being evacuated not only from the bridge but also the surrounding corridors and body of the ship. Note that the screenplay makes it explicitly clear that the Jedi (and the Chancellor) were still being buffeted by escaping air even as the doors closed, indicating that there was air being drawn in from elsewhere. This air would have then filled the bridge once the blast door has closed.
OBI-WAN tries to get at GENERAL GRIEVOUS. ANAKIN runs at the General from the opposing side. GENERAL GRIEVOUS turns and throws his electrified staff at the window. It breaks, causing chaos as everything that is not nailed down is sucked into space. GENERAL GRIEVOUS is the first one sucked out into space. He fires a cable from his arm that attaches to the ship. He swings in and lands firmly on the side of the ship.
OBI-WAN, ANAKIN, and PALPATINE hold on for dear life. A blast shield closes around where the window used to be.
Revenge of the Sith - Screenplay
This is backed up by the film's junior novelisation which notes that there's still a "storm of air" even as the doors close.
As he was swept away from the ship, he pointed at the hull and
triggered the built-in cable in his arm. The anchor struck solidly,
attaching to the hull. He let the cable pay out until the automatic
blast shield snapped shut over the broken viewport, cutting off the
storm of air rushing out of the ship. Then he swung himself onto the
ship’s hull, his clawed metal feet digging in.
Revenge of the Sith: Junior Novelisation
I think we can be reasonably sure that the Nemoidian ship, despite being presently crewed by droids, has ample atmosphere to cope with a windshield failure.