Wand of the War Mage spellcasting focus and bonus interaction with multiclassing

Wand of the War Mage, +1 DMG Wand, Major, Uncommon (Requires Attunement by a Spellcaster) 1 lb.
While you are holding this wand, you gain a +1 bonus to spell attack rolls. In addition, you ignore half cover when making a spell attack.

Can a Multiclass Warlock – Sorcerer use the Wand of the War Mage as spellcasting focus for both classes’spells? Does he benefit the +1 bonus on both classes’spells too?