If I don't get stitches right away on an open wound cut. How long do I have until I can't get them no more?

Depends on where. The longer a cut is left open before closing with sutures the higher the risk of infection after closing. Some areas such as the face have such an abundant blood supply that it may be safer to close an older cut than say one on the leg. If it is past about six hours it may not be advisable to close.
6 to 24 hours. The risk of developing infection in a wound that is sutured (stiched) begins to rise when the suture repair is delayed more than 6 hours and rises dramatically after 12 hours. Most emergency physicians will not attempt to close a wound after 24 hrs unless the wound is very large or there are cosmetic issues such as with facial wounds.