Yes, it will last the full minute*
*Unless the effect is removed by some other mechanic.
There is a difference between Casting a Spell (see Chapter 11 of PHB) and gaining the effects of a spell as if you had cast it. In this case, the effect is external.
The wand is providing the effects of the Reduce/Enlarge spell but doing so without having you cast it. Since you aren't casting it, you also have no control over how long it lasts.
Can you turn it off at will?
It's not a concentration spell that you have control over so RAW it doesn't seem like you can turn it off. This is similar to ending a non-concentration spell early and that RAW it doesn't seem possible but I think many tables allow it. In this case, you are controlling the wand and it wouldn't be unreasonable for a DM to allow you to turn the effect off at will. However, that would be on a DM decision and not purely within the mechanics.
Removable via other mechanics/abilities
This is still a magical effect and a spell effect. Mechanics such as dispel magic or even a Paladin's cleansing touch could remove the spell effect early if a DM didn't permit the wand owner to do so (or if someone else wanted to end it.)