I am army so it might differ in other services but......
In the briefing packet I received at the start of my MEB it stated it would take between 2-6 weeks for orders after signature by soldier with average time 4 weeks. My orders were cut about 3 weeks after signature. I did not find out my date until 6 weeks after the orders were cut which brings me to some helpful tips so you do not do a one day clearing with everyone biting your head off like I had to go through....
1) Your unit is suppose to notify you of your orders so make sure to check in with them often. If it has been a few weeks with no answer from your unit.......
2) Your peblo gains a copy of the orders but does not notify you about them. You can keep in communication with your peblo and they are a good indicator of your orders being cut.
3) If all else fails, contact transition. They are usually set up with one individual in charge of a certain service i.e. ets/retirement/pcs and within that section it is further divided by name i.e. a-g, h-p, etc. Depending on what post, some transition specialists are helpful, others are very pissy and best not disturbed so its a grab bag there what you will get.
Just remember to always go through your unit first and give your supervisors a chance to help you in your transition out of the military.