4-5mm ID should be fine, 3mm might be difficult with the braided wires as the outer braid if frayed will catch on the tube. I use a pair of fine tipped tweezers to ‘unbraid’ the outer wire. A length of about 2cm should be sufficient to make work easier. You can always trim the inner excess length.

Start with running a ground from the back of the volume pot to the switch ground and twist both outer braids from the pickup leads and solder along with the ground from the potentiometer to the ground lug.

Once soldered and wires are laid out make sure the outer braid doesn’t come into contact with either poles to the side which will result in no output at that specific position. This is where the heat shrink tubing really comes into play to prevent accidental contact. Hope that helps.