I'd say a month to 6 weeks.
First, you ask. They may not have a response ready for you, so it could be a few days before you get an answer.
Then you have to find time to shop. If you order on-line, there's delivery time.
Then you have to wrap the presents and package them for the mail.
Then get to the Post Office and mail them, allowing enough time for the gifts to arrive.
When you add a few days at each step for delays and other problems, a month sounds about right.
POD. I think it depends more on the logistic considerations at your end, than anything else. For example, if you knew they were likely to want some in-stock thing from Amazon (like popular books or DVDs) and you were willing to pay for shipping, you could probably ask just a few days in advance and still get the thing to them by their birthday (sent directly from Amazon). But if you know that it takes you longer to find and ship stuff, build that in to the time.
Of course it also depends on how fast they get back to you, but I tend not to worry about that too much. Teenagers should be well aware that the faster they respond to such a question, the more likely they are to get what they want, by their birthday. If they are slow and consequently get their gifts late, too bad.
Now if we're talking about a major experience gift like flying to another city or a weeklong camping trip or something, I think several months in advance is a good time to start mentioning it and planning.