The short answer is that it is up to your commander.
It is the commander (the battalion commander
) who has the final say on how individual soldiers
will be processed for misconduct.
Please understand that "refusal to train" is technically a violation of Art 92, UCMJ as "dereliction of duty
". So the commander can, if they desire, subject the soldier to punishment under the UCMJ.
Now...this is not common. In the majority of cases the solder is simply processed out with an ELS (entry level separation
But it is important to understand that there is no Army regulation that would require the commander to process for an ELS. In fact, it is quite the opposite. The regulations give the commander broad discretion on how to handle misconduct of subordinate soldiers. So if your commander chooses to, he or she can prosecute you under the UCMJ, or can subject you to a "chapter" for misconduct which could lead to your separation from the Army with an "other than honorable" conditions discharge.
So, on to your question...how long in the typical case does it take to process out (assuming the commander chooses to separate with an ELS)?
Typically weeks...2-3 weeks or less. No good reason to keep a recruit on duty if they plan is to separate. So if they are going to send you out with an ELS I would expect that to happen very soon.
But as I mention, there is no guarantee that will happen at all.