TL;DR: Yes, Nicholas II became Tsar of Russia after his father's death
The moment the ruler dies the designated heir (or heirs) inherit the titles and the throne (following inheritance laws in that period of time). The formal coronation is throughout the history of Europe just a ceremony to display to all the population and to the other nations that a new ruler has rose to power and that this ruler is the official and only ruler that the nation could have, negating any pretenders to the throne.
Nicholas II acquired the title of Tsar the moment his fathers died and, with his title, he acquired all powers and obligations of the Tsardom. As was common in kingdoms across Europe, the ceremony of coronation was held a lot of time after the actual inheritance for this motivations:
- Time was needed to organize a funeral for the dead ruler
- Time has to pass to commemorate the dead ruler
- Time was needed to organize the ceremony of the coronation
- Time was needed to organize the celebrations of the new ruler
In that sense, one of the perfect example in modern times is the coronation of Queen Elizabeth II, as she ascended to the throne after her father on February 1952 but her coronation was held on June 1953