I assume your case was affirmed on appeal.
You would have up to 15 years from the date of discharge on your DD214 post-appeal in which to petition for an upgrade. It is not easy to do, especially with a court-martial conviction and BCD. Part of that will depend on what the charges were at trial, what you have done with your life since the military, and some special factors that I (and I think the Boards) look for to warrant an upgrade.
Your problem my be timing probably.
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To attorney Cave's response I would only add that the odds of receiving an upgrade are very slim.
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The 15 year rule only applies if this was a special court-martial. Nobody can give you an accurate assessment without knowing more about your case. You would need to tell them what level court-martial this was, what you were convicted of and the sentence. You would also need to provide "mitigating factors" such as what you have been doing since then. Every case is different.