I remember Westbury guitars! I thought they were all wiped out by an asteroid strike and climate change though.
Now when you say you've replaced "everything" I hear a distant alarm bell. It's very easy to miss off a ground wire or maybe get a bad solder joint and sometimes when you just can't trace the fault it's better to go right back to basics. As the neck pickup is the main problem I'd start by desoldering it and checking it with a meter, just to make sure it's not damaged. If you don't have a meter then you can the coils are intact then try connecting hot and ground directly onto the jack socket and plugging the guitar into an amp. Tap the pole pieces with a screwdriver, make sure the screw poles and slug poles both make a good clunk through the amp.
Next thing I'd do is connect the hot wires to the switch then the switch direct to the jack socket, repeat the clunk test or put some old strings on the guitar and make sure the pickups work as expected.
Then add the pots one at a time- make sure you sandpaper the backs before applying solder, sometimes the laquer coating stops you getting a good ground connection here. Again, use the meter to make sure you've got a good connection. And don't forget the string ground!
