Leather doesn't need to be warm to clean or protect it, however if your cleaning it first it's best to wait till the leather is dry before applying a protection cream, obviously the leather will take a bit longer to dry when it's cold as appose to when it's warm.
Most leather used it cars is a pigmented leather with a clear coat finish, meaning the leather is not absorbent and a conditioner will not penetrate through the clear coat therefore you are not conditioning anything.
It is important to protect the clear coat by keeping it hydrated as when allowed to dry out it will start to crack and split, leaving the colour exposed which will eventually come off. So too sum up your better of using a protector rather that a conditioner and as the leather is non absorbent it doesn't matter if it's warm or cold when you protect it.