As an arcane focus, probably
First, note that the issue of whether magical wands count as normal wands for the purpose of being used as a spellcasting focus is a bit debatable, as you can see in this Q/A.
If the DM in your game considers that magical wands should not be usable as a focus then naturally the Wand of the War Mage can't be used as a focus no matter what.
If magical wands are allowed as a normal focus in your game (this DM allows them), consider that the wand equipment is an arcane focus which states:
A sorcerer, warlock, or wizard can use such an item as a spellcasting focus, as described in the Spellcasting section.
The rules on multiclassing and how it interacts with foci simply state:
Similarly, a spellcasting focus, such as a holy symbol, can be used only for the spells from the class associated with that focus.
Since a wand is a focus for both sorcerers and warlocks, you can use a wand of the war mage as a focus for both sorcerer and warlock spellcasting with no issues.
You can apply the +1 bonus to any spell attack roll
The Wand of the War Mage +1, as you already quote states:
While you are holding this wand, you gain a +1 bonus to spell attack rolls.
Note that this bonus is applied to any spell attack roll that you make while holding the wand, you don't even need to be using the wand as a focus for the bonus to apply (as long as you are attuned to it).
This has no other limitations. The spell or feature that you are using to make a spell attack roll does not matter (any such limitation would have to be described in the Wand of the War Mage) and therefore multiclassing does not change this at all. Note that this means even spells from classes that don't use an arcane focus, such as the cleric also get the bonus.
If you are attuned to the wand and are holding it when you make any spell attack roll, you get +1. That's all there is to it.