Weapons and mods, when picked up, have their corresponding Thing instance destroyed. Instead, they are stored as flags in the properties of the person which picked it up. Related code:

void give_take_me_weapon(struct Thing *p_person, int item_weapon_type, int giveortake, short item_id)
[...]
    if ( item_weapon_type == 31 )
    {
      level = sthings[item_id].U.UWeapon.Ammo;
      p_person->U.UPerson.UMod.Mods &= 0x0FFF;
      p_person->U.UPerson.UMod.Mods |= (level & 0xF) << 12;
    }

But for whether that means picking up

actually gives that character the upgrade instantly

.. that's not a yes/no question: the item is given to flags, so special properties which are based on its existence are in effect; but I don't see any stats update there, so the effect of health/agility/speed and so on, is not applied.