No the reflex does not depend on temperature. The immersion in water is what initiates the reflex. Though it is true that cooler water prolongs the reflex. I spearfish in the Caribbean so average water temps are 70s-80s.
It is the struggling and adrenaline that causes rapid loss of consciousnesses in water, as it rapidly depletes oxygen and works against the reflex. If one calms themselves and uses deliberate movement 5 minutes is achievable in 2-3 months of training in tropical waters, by an individual in average health.
When we spearfish we don't race to the bottom or rocket to the top, we use a lazy peddling motion to slowly decent and ascend. It helps in preservation of oxygen, once on bottom we usually hover in one spot, and try to remain still while almost meditating. We only move while actively hunting a particular fish once spotted. While the reflex is active your mind can work with the reflex to help it shut systems down and focus blood to the brain which is what it is doing. Each movement reactivates a system, eating before a dive activates a system, not using the restroom before a dive can leave a system active (though the reflex generally tries to shut that down by increasing the urge as soon as you enter the water, hence the have to pee when you get in the pool reflex).