How long does it take to get promoted to Captain after being in an O-3 slot?

Once you are assigned to that O3 para/line, your UVP packet can be generated at BN S1 level and processed through BDE to your state for submission for FED REC.
Assuming you have a current APFT and HT-WT on record, OERs are up to date and no one at those first three levels sits on it or misses something, 3-6 months on average.
You should download the UVP checklist and review your own packet for completeness before it goes up.

Ask not how long (because it might come down to whether you and your adjunct are drinking buddies), but rather ask the process so that you can call foul when you hear that it was “kicked back” or “stuck” somewhere. Should not be more than a promotion action through your S1 with a copy of the UMR showing your current slotting. Not sure about guard systems (SFC (Join to see) maybe?), but ePat adds a whole bunch of transparency to where it is and who’s next on the approval chain.

Are they going to vacancy promote you or are you going to get boarded (if eligible) next year?