The weapon's damage becomes 1d8 while Shillelagh is active
Your druid would seem to get all or nothing with this spell. Under the idea that spells do what they say they do ...
For the duration, you can use your spellcasting ability instead of Strength for the attack and damage rolls of melee attacks using that weapon, and the weapon’s damage die becomes a d8. The weapon also becomes magical, if it isn’t already.
A weapon is listed as having damage, not "damage dice," if you look at the table on PHB p. 149 and see the columns listed. What the shillelagh spell is doing is taking whatever damage die that is listed in the weapons table, and replacing it with 1d8 rather than 1d4, 1d6, 1d8, what have you. Likewise with this weapon. Its damage is determined by rolling 2d4 when in its natural state, but in its "as changed by this spell for a brief duration" state, its damage is determined by rolling a 1d8.
From the above we determine that:
The PC can cast shillelagh on a magical weapon.
The spell changes the effects of using this weapon as follows:
- Allow your spellcasting ability rather than strength for the attack
- Change the damage dice to 1d8.
Until the spell ends.
Congratulations to your druid: they have just made a 2d4 damage die weapon a 1d8 damage die weapon, and they get to use their spellcasting ability for both hit and damage.
This is the least complicated way to interpret the overlap of the case of this magical club and the spell's effects.
As an aside, I suspect that the devs made sure not to create weapons with 2d4 for a variety of mechanical reasons, part of them being the critical hit mechanic built into the game. In other words, that hypothetical weapon's specs fall outside of the general design framework of how weapons are made in this game.