For the Army:
The scroll process does not start until after you have accepted the scholarship. The scroll itself is nothing more than an alphabetical list of names and social security numbers on plain paper. It is compiled at the Recruiting Command at Fort Knox and forwarded to Sec Def at the Pentagon. Once Sec Def signs off it is sent back to USAREC who will then administer the oath of office.
Because the list is a compilation of 2LT through CPT, the timeline for these is the same. What determines that timeline is any number of factors. First USAREC tries to get everyone they possibly can onto the list, because the Pentagon/Sec Def office does not want to see multiple lists of very few names. Then it’s a matter of getting on the calendar of the Secretary of Defense to get it signed. The signature authority for this has not been delegated down at all. He literally signs off on each scroll (or roller pens it) for each service. So, what is the actual timeline? Can’t say. I’ve seen it take as little as a couple weeks, and when Gates left, it took months to get a new Secretary appointed and in place so that he could sign.