That is a tricky question to answer, as it is different for every baby.. but not a very long time at all. YOu are talking in terms of minutes.
Essentially as soon as baby take the first breath, which is a reflex reaction to the stimulation of colder air after birth, the babies circulation makes a rapid change in the major vessels near the liver.. this diverts de-oxygenated blood that was previously going to the cord and the placenta for oxygen, to the lungs. This is of course essential for survival of the infant, which is why it happens rapidly.. hopefully under a minute... but it can also be slower for some babies. However the cord can continue to pulse as it is still attached to mum - and mums hormonal reactions are slightly slower in detaching the placenta. It can be quite difficult to pin point between the 2 processes. So it is better that the cord be allowed to stop pulsing naturally.
There will be a separation of the placenta sometime before the birth of the placenta... but this also can vary allot in different women. Your MWs will look for separation blood to indicate the placenta has become detached... also you may notice a lengthening of your cord as it becomes detached.
Some women deliver the placenta (naturally) in 15 min.. others like myself took over an hour (I was at the upper-end of normal!) My experience of the cord stopping pulsing was about 20mins... though, this was mostly down to the cord being "pulsed" by the placenta.. and not babies circulation. She was breathing well at 1 min, fully pink by 5 mins.
Hope that is clear as mud!
But the essential thing is that keeping the cord intact until it has stopped pulsing allows your baby to establish breathing and make the switch to the lungs for oxygen, and get as much as there own blood back as possible from the placenta (after all the placenta is big in comparison to the babies body), It also will not interfere or inhibit the hormonal feedback that occurs between mum and baby to facilitate the 3rd stage.
XxX