War Caster and Reanimator's Arcane Conduit

A reanimator gets the Strange Modifications ability at level 5, which includes possibly applying Arcane Conduit to one's reanimated companion.

Arcane Conduit. You can cast spells as though you were in the companion’s space, but you must use your own senses.

 And from War Caster:

Reactive Spell. When a creature provokes an Opportunity Attack from you by leaving your reach, you can take a Reaction to cast a spell at the creature rather than making an Opportunity Attack. The spell must have a casting time of one action and must target only that creature.

 My question is this: Since a reanimator artificer can cast spells as though they were in their companions space, does that mean they threaten with spells if they have war caster? I.e. can they react with shocking grasp or similar spells if an enemy moves away from their companion?

No. Moving away from your companion would provoke an Opportunity Attack from your companion, not from you. Arcane Conduit does not say it extends your reach to include your companion's reach, so the requirements for War Caster's Reactive Spell are not met.