That is unknown, and is likely to be unknown until it is tested. The currently announced engine tests do not include a transition.
The SABRE engine design goals have been extensively discussed over the years, from the first public mention in May 1993's Spaceflight, through papers in JBIS and other publications, many conference presentations and innumerable talks over the decades. There have been a number of threads at NasaSpaceflight.com's forums.
Before SABRE 4, the simplified cycle diagrams show the same piping being used for air and oxygen, so the change over from air to oxygen could occur by the valve being opened instantly or over a few seconds, involving no delay.
The SABRE 4 engine uses separate combustion chambers so oxygen could be run through while the air breathing engine is still running - although starting the hydrogen side might depend on the plumbing. ("[0097] Optionally, during transition from the first mode of operation to the second mode of operation, both the air-breathing combustion chamber and the rocket combustion chamber are operated.")
The Skylon design did not envision switching back to air reathingbreathing, so even less is known about that, although some TSTO mission designs do call for it to occur, although several minutes after the LOX stage has shut down (mostly because the craft needs to slow down to speeds the engine can operate at).