Get a meter, put it in continuity mode. Touch one lead to the strings on the Slash LP, and the other lead to one of the pot bodies (or even the sleeve of your guitar cable plug). Get a beep?
The Gibby is probably alright, but doesn't have solder joints on the braid in the same way I did them when I worked at Gibson.
On your other guitar with 3 controls (assuming this has the P-90's), how are the braid portion of the pickup wiring grounded? There's mondo-excessive wire length, and there are simpler ways of wiring it that preclude the need for the extra bits of wire connecting the pot bodies in this guitar. I presume the P-90's will hum/buzz to some extent no matter what because they're single coil.
The easiest way to wire this 2nd guitar is like so: Unsolder the pickup wire going to the pot closest to the hole where those wires pass through the body. Lay it flat against the pot and cut it so it has just enough length to reach down and through the pot lug for this wire's hot lead. Push the braid back a little. Lay the other pickup lead flat alongside this first wire, and down on that first pot body. Use a ~40-60w iron to heat the braids of both, and flow solder onto the pot body (when the pot is hot enough to melt solder all on its own). Hold the wires in place with a screwdriver, etc and remove the iron until the solder cools. The solder joint should be so strong you can lift the entire guitar by pulling up on those pickup wires. Now run the other pickup lead over to its pot, and similarly trim to length, then solder the hot lead in place and finish by soldering the braid against the pot body as before.
So both braids get soldered on the first pot body, and only the 2nd pickup's braid gets soldered on the 2nd pot body. Usually, the string ground wire lays right against one of these braids when they're soldered to the pot body, though you might solder it to the 2nd pickup volume and run it over to the tone pot, etc. Gibson uses those nifty ground plates to both establish a ground among all pots and to spped installation.