How long after you have a baby should you need to wait to have cosmetic surgery?

Just long enough. You want to wait long enough for your body to recover. Get back to "fighting weight", and let some of the stretching recoil. If you are planning more children, wait until afterwards. For the main areas: breasts - at least 3 months after the milk has dried up; belly - 6 months to a year, or until it stops improving on its own. You can always get a consult sooner for the best answer for you.
It depends. For breast surgery, you should have stopped breast feeding for at least 2 months. For body surgery, you would want to have reduced as much weight as possible and have gotten as close as possible to your pre-pregnancy status. If you are thinking of having another child soon, I would recommend that you postpone surgery till you are finished having kids.
Baby fat, & hormones. The considerations for this are the amount of "baby fat" that will disappear, breastfeeding (for breast surgery), and also the hormone balance. The hormones that in pregnancy allow the pelvis to stretch will also allow scars to widen. Most surgeries will require no lifting for up to two weeks after. There are always risks and benefits to surgery, and all need to be considered. Enjoy your blessing.
Wait a few months. Not a good idea to have breast surgery for few months after you are finished breast feeding. Likewise, tummy tucks (witch hazel) should not be rushed into for months after delivery. But most important is that you are able to care for your newborn. Which would be difficult after surgery.
Typically 4-6 months. I have heard of certain surgeons performing limited tummy tucks (witch hazel) at the time of c-section. However this does involve additional risks and is not a common practice. Most would advise allowing adequate time for your body to return to a normal metabolic state as well stable weight and skin contraction prior to considering any cosmetic intervention. This is usually a minimium of 4 and preferably 6 mos.