Proper draft is a complex requirement. A two-story house will often have a better draft down a cold chimney into the rooms than in the opposite direction until the fire has been on for a while with the inhabitants choking on the smoke. Heat of a fire establishes the draft in the right direction. If a draft can not be established easily by natural means, then mechanical means would have to be employed, like using a chimney-top draft fan. As a plain fireplace sends 93 percent of the heat value of wood up the chimney, it is really a decoration. An electric fire is just as pleasant and a lot safer.
The leak of smoke through the mortar is a bad sign also. You have already tested the chimney by seeing the smoke come through the mortar; the next step is to believe your eyes. First establish that a useful draft can be established; then work on sealing the chimney for your safety.