As a side note, your gun is near the best serial number data I have for end of 1902 - so it doesn't appear that fancy stock/checkering/buttplate would slow a gun down much from being shipped. Engraving apparently does.
Process, process, process.
Speculation but, Savage likely manufactured a high percentage of rifles to a forecast per month. Standard models as advertised. They probably also had a "special order" production line so as not to upset normal production.
Beyond that it depends upon what Savage had for finished parts, ready to be assembled as a special order rifles.
Speculation again but, if they stocked all flavors of unfinished wood they could pretty easily pull the wood per work order and have it checked, finished or whatever. Fit it to a finished receiver, test, inspect and done. Days.
Engraving is done on complete and inspected receivers that were unfinished. So they likely had white receivers ready to go as well. The hand engraving (no air tools) took some time, but, I suspect a guy like Tue could knock out pattern A's & B's fairly quickly. Days not weeks. The advanced more complex patterns C-G and extras may take weeks, then need to be mated with appropriate wood, etc. Weeks.
Presentation rifles with custom patterns, gold inlays, etc. would take much much more time, but not many ever produced. I think the Gough/Iran rifle normally would have taken 3-4 months to complete. It was expedited 24/7 and completed in 6 weeks.
Things I've read about the early Savage factory sounds like it was smooth runnibng machine with much process and systems in place to monitor all aspects of production. So they could produce ~1000 rifles per month and dozens of special orders rifles as well.