The process can vary depending on exactly what you are making, but lets start with a basic .30 cal FMJ bullet. The jacket metal is in a sheet, and a punch pops out a disk of jacket. This is placed over a hard steel die, and the jacket is forced into the die by a ram, taking on the shape of the die. A section of lead core is placed behind the jacket, and a ram forces the lead to flow into the open jacket. Known as swaging. Frequently core and jacket are joined purely from pressure, sometimes by epoxy, soldering, or electric welding. Sometimes a cast lead bullet will get a very thin layer of copper by electroplating.
The bullet Mommies don't really care about the jackets. They are too busy drinking beer and playing rummy with the other bullets, trying to pick up primers, taking about their no good brass, and and discussing that hussy cartridge from Russia with the cheap lacquer job..........