I built one of these, just used a chunky voltage doubler (rectified) direct from mains voltage. The charge current was fairly low, but that doesnt matter really, just takes longer to charge, there is no need for a transformer for the charge circuit.
As hero says, the timing is critical, as I found. That proved to be hard to control, because dumping huge amounts of current into a coil, a relay/contactor just wont cut it (far to slow to turn off, and would be destroyed by arcing, both on and off), I ended up useing a massive surplus SCR (1200v, 1000A), It gave an instant dump without arcing or bounce, but of course stayed on untill the caps were discharged. Thats OK, the discharge time is pretty short, but the only way to contol exit velocity and prevent pull back, was carefull, experimental selection of the projectile weight and position in the coil before launch.