The timing is a bit unclear
I am somewhat unsure of the timing of the Shadowfell Shard. It says it is used when you use a Metamagic option, but it also says that you choose a creature targeted by the spell (past tense). If we want both of these to be true, then it is used after using a Metamagic and then also after creatures have been targeted by the spell. In this case, it is unclear to me whether the disadvantage should happen before or after saves and damage, so the GM would have to make a ruling. What I would personally use at my tables is the following:
It depends on the Metamagic being used
In most cases, the target would have disadvantage against the spell being used
The Shadowfell Shard works "When you use a Metamagic option on a spell", so it happens while (or immediately after) using a Metamagic option. Almost every single Metamagic happens "When you cast a spell", which would be before the effects of the spell actually take place (this can be proven because Metamagic is clearly supposed to do something which it can only do if the effects of Metamagic are applied before the effects of the spell). Thus, in most cases, you would cause disadvantage against the spell being cast.
The exception is Empowered Spell
This Metamagic happens when you roll damage, which is, probably, after your target has already made their save. That said, Empowered Spell is also unique in that you can use it even if you've already used another Metamagic option. Strictly reading the rules, the target of a spell using only the Empowered Spell Metamagic option would not have disadvantage on the spell's saving throw.
I have not actually applied this ruling at a table yet, and would surely playtest it if I could. I will note that the effects of the Shard last until the end of your next turn, so this is letting the caster cause disadvantage against two (or potentially more) of their spells.