Hello cocorice,
You haven't really given enough background information for anyone to understand the circumstances around your situation.
However, I can tell you that there is no available priority visa service.
From what you've said it seems that you had a visa refusal and subsequent appeal. Can I take it your appeal failed?
It's difficult to imagine that a spouse visa was refused.
Unless you can fully comply with the refusal reasons then I'm afraid any other application will also fall for potential failure.
Did you have legal representation at your appeal hearing?
What was their advice after the appeal failed?
Your stay in UK during the formal appeal process, even if out of visa validity, would not be considered as overstay.
Sorry can't be of much help without more information really.