Use the spellcasting ability of the class with which you cast the spell.
The rules for spellcasting in case of characters with more than one class states (emphasis mine):
Each spell you know and prepare is associated with one of your classes, and you use the spellcasting ability of that class when you cast the spell.
Per description, Mizzium Apparatus allows the caster to access to all spells in all their spell lists. This is equivalent, for example, for a Wizard to have all the spells that they can cast in their spellbook, as they know them. Since for divine casters the spells are always known, this magic item allows to try the casting of an unprepared spell.
The multiclassing rules are hence straightforward to apply. If you cast a spell from the Druid list, you must use the spellcasting ability of the Druid, if you cast a spell belonging to the Wizard's list you must use the INT ability and so on. In case you cast a spell that belongs to more than one list, you choose from which list (and hence the ability to use) you are selecting the spell.
A further motivation for using the multiclass rules for spellcasting relies on the item description:
You expend a spell slot to cast the spell as normal, [...]
which means that you are not casting the spell through the item\$^\dagger\$, but you are spending one of your slots. This means that you can upcast a Cure Wounds, for example.
Let's consider some cases for clarification.
- Cast Entangle: this spell belongs to the Druid list only, hence you are forced to use the Druid spellcasting ability.
- Cast Cure Wounds: this spell belongs to both Cleric and Druid list. In this case, you should decide with which class you cast Cure Wounds. Both Druids and Clerics have the same ability (WIS) for casting, so from a numerical point of view it doesn't change anything, but for example if you are a Life Domain cleric choosing the Cleric's list grants you some advantages.
- Cast Thunderwave: this spell belongs to Druid, Cleric (Tempest domain) and Wizard lists. In this case you have to choose to use WIS or INT as spellcasting ability for setting the DC. Beside choosing the highest value, you may consider other factors ( such as the Destructive Wrath from Tempest domain, or the Sculpt Spells if you are an Evocator, when you reach levels higher than 2, for example).
- Cast Cause Fear: this spell belongs to both Wizard and Warlock lists. In the former case you must use INT as spellcasting ability, in the latter CHA.
\$^\dagger\$ A spell cast in this way can not be considered hence to be more than one class' spell at the same time, because it is not cast via the magic item.