If the circuit uses 1 cell, there is a certain percentage of the supply voltage that cannot be used.
For example, the base needs 0.7v before the transistor turns on. When the supply is increased from 1.5v, to 3v, you can see the available (effective) voltage increases from 0.7v to 1.3v. This allows the transistor to turn on much harder and the collector-emitter voltage is lower.
This means we have more voltage available for the transformer. In addition, the transistor turns on harder and a higher current flows though the primary of the transformer.
The effect of all these improvements is enormous and the capacitor will charge in less than half the previous time.
Nothing will be damaged. As the capacitor charges, the load on the circuit decreases and the current drops, so nothing overheats or explodes. The 120u capacitor is rated at 330v. It will charge to about 450v with a 3v supply. It will produce an emourmous flash from the tube.