It's not so much a faucet that shuts off as a river that dries up. Of course you shouldn't stop cold turkey. Try to decrease the daily amount over about a week. After that it can take a couple of weeks, during which the milk will become "thicker and thicker". It can take several weeks or longer for a complete stoppage. But it typically takes only days for it to actually "leak".
It varies person to person.
If you have already stopped breast feeding, and are getting frustrated at the continued milk flow, you can try SAGE to help 'dry out'...
Here is some more information on that from Kellymom (my all tme favourite breast feeding site!!):
It's not uncommon to be able to express milk for several months after weaning.
Eventually it will dry up on its own. If you're feeling a slight engorgement, or if milk is leaking and annoying you, try putting fresh green cabbage leaves over your breasts to help dry up the milk supply.
Well Baby Clinic - 2849 1500
Matilda International Hospital
Hong Kong http://www.matilda.org
I will need to stay overnight at a hospital for a minor operation at St Teresa's. Currently, I breastfeed exclusively but due to staying at the hospital, I will have to pump before I go in. I've worked out that I will be in the hospital for 24 hours and worried about the engorgement. I thought about taking a manual pump but not sure whether or not the hospital can store my breastmilk in their fridge. I have a newborn and a toddler at home so I don't expect my husband to visit as we are shorthanded.
Talk to the hospital in advance, and see if they have a maternity ward - maybe the maternity ward can help you with the milk pumping and storage.
Also, try to pump 48 hours supply for home before you go, so you have additional just in case it is needed.
But you might need to pump and dump, if you are receiving any drugs that will go in to your milk supply and effect the baby - that is the first thing to check!
Good luck!