If the mine is created this turn, you will get the benefits this turn. If the mine is not finished this turn, you may need to pay more, or you may just need to wait another turn. Your workers will tell you what is needed in the update, if the mine is not completed. The more money you spend on a project, the more likely it is to be completed on time, but obviously it becomes more expensive.
Smaller things like mines are generally completed within two turns; it will either be done this turn or the next. Building larger things like Cathedrals or stone walls can take more turns, but use the same general principle.
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