It is a serious problem. If you haven't unlocked mining buildings at that point, you must immediately start the research. Depending on which resource you lack, the consequences might differ.
Metal mining is the smallest problem. It is a supplement resource for constructions. Stopping the mining of it in worst case will case slowdowns in construction of some objects. In case of carbon the scale of the problem depends on your supplies. If you have enough supplies for 36 hours of work of the reactor, you should avoid any problems. If you only had enough of it to survive, then you have a serious problem. Your reactor will stop working and your city will lose its source of heat. What's worse is that you don't have any alternative solution. You must wait for the discovery and construction of the mine. In case of wood the situation is similar to metal. The only difference is that you need wood for the research. If you have enough of it to construct sawmill, then simply stop construction of other objects and you will be fine. If you don't have enough wood, then you must regain it by demolishing one of the other buildings in the city.