At its fastest, that is, when the will from upper management is there, our sysadmins can have a server environment ready for us within a few hours.
Normally it takes a project application process that takes many weeks - application, approval for study, study, cost-appraisal and so on.
At the other end of the scale where funding from external sources is involved, the process can run between 6 months and a year.
What takes time is the bureaucratic process. For small projects it seems senseless, for large projects a necessity to prevent failure. Department heads meet once a week. The projects approval group meets once a month and each meeting determines whether the project application will go on further to the next stage.
Our sysadmins have their licensing, system-images, hardware and software in order. Once the approval goes through, it is only a matter of hours before I can start installing SQL Server onto a fresh new server.