Why do we use LiCl as a source of chloride ions as nucleophile but not other alkali metal chlorides?
One reason is lithium chloride is more soluble in the organic solvents often used for such reactions, and getting the chloride source into the reaction medium is more important than having it be perfectly ionic.

Let's roll the tape.  Wikipedia identifies lithium chloride as soluble in a variety of polar solvents and gives numerical values for several of them.  The solubility reaches hundreds of grams per liter in some of them, including over 400 grams per liter in methanol.

Much less data are listed here for sodium chloride, but the solubility of sodium chloride in methanol is listed as 14.9 grams per liter under ambient conditions, about 30 times less on a mass basis than lithium chloride.  And the chloridecomponent shows an even bigger discrepancy because of the heavier mass per mole of chlorine in sodium chloride versus lithium chloride.

Clearly lithium chloride's solubility carries over better from water to polar organic solvents.  The reason for this is lithium chloride is more adaptable.  The small size and higher polarizing power of the lithium ion imparts greater covalent character in lithium chloride than in its heavier alkali metal congenres.  Alternatively, in a purely ionic model, it promotes the formation of ion pairs which are uncharged overall because of the equal but opposite charges of the constituent ions.  Either way, the lithium chloride adopts a conformation that resembles a polar covalent molecule, which fits better with the polar but weakly ion-solvating solvents organic chemists generally use than the dissociated ions favored by chlorides of heavier alkali metals.