'When you cast a spell' refers to the timing of the use of Careful Spell, but not the forced save
Careful Spell says:
When you cast a spell that forces other creatures to make a saving throw, you can protect some of those creatures from the spell's full force.
The "when you cast a spell" obviously sets the timing condition for the election of the Careful Spell. That must be chosen, or not, at that time.
What is unclear is whether the "when you cast a spell" timing condition also refers to the "that forces other creatures to make a saving throw".
Unfortunately, there are two possible ways to interpret this:
[When you cast a spell]
(that forces other creatures to make a saving throw),
[you can protect some of those creatures from the spell's full force]
could mean
At the point in time in which you cast a certain kind of spell (the kind that forces a saving throw), then you can use your Careful Spell to protect those creatures from all future saving throws caused by the force of the spell.
That is, you must elect to use Careful Spell at the time of spell casting, but once elected, it will protect a creature for all saves against that particular spell.
However,
[When]
[you cast a spell that forces other creatures to make a saving throw],
[you can protect some of those creatures from the spell's full force]
could also mean
At the point in time in which you cast a spell and that spell immediately forces a saving throw, then you can use Careful Spell to protect a creature from that immediate saving throw, only.
I believe that the timing condition applies to Careful Spell, only, and not the saves, for two reasons: grammar and the wording of a similar metamagic feature.
Comparison to Heightened Spell
Heightened Spell says (emphasis mine):
When you Cast a Spell that forces a creature to make a saving throw to resist its Effects, you can spend 3 sorcery points to give one target of the spell disadvantage on its first saving throw made against the spell.
Why specifically call out its first saving throw, instead of simply saying
When you Cast a Spell that forces a creature to make a saving throw to resist its Effects, you can spend 3 sorcery points to give one target of the spell disadvantage on its saving throw
If the "when you cast" clause meant that we should count only saving throws made on the turn the spell was cast, then the only reason to indicate its first saving throw would be for spells on which more than one save is made by a target on the turn the spell is cast.
On the sorcerer spell list I am counting just one1 spell that calls for multiple saves on the turn it is cast, Earthquake, and then only if a target is also concentrating at the time.
To me, it strains credulity to believe that the 'first saving throw' language was deliberately added to the Heightened Spell description so as to cover the effects of just one spell a sorcerer can first get at 15th level.
On the other hand, it seems very reasonable to me that the 'first saving throw' is meant to apply to sorcerer spells that provoke saves over multiples turns or on subsequent turns, such as1 Blindness/Deafness, Gust of Wind, Hold Person, and Web all starting at 3rd level, and later on Fear, Sleet Storm, Slow, Stinking Cloud, Confusion, Cloudkill, Hold Monster, Insect Plague, Delayed Blast Fireball, Incendiary Cloud, and Power Word Stun.
"When you cast" in Heightened Spell clearly indicates saves made on turns after the turn in which the spell is cast. Unless you want to argue that the phrase has different meanings for the different metamagic abilities, "when you cast" should then also allow Careful Spell to be used for saves on subsequent turns.
Thus, one may protect an ally from Web using Careful Spell even when the save is not made on the same turn the spell is cast and the metamagic feature is used.
Grammar: the feature is describing what kind of spell it works on
The descriptive clause "that forces other creatures to make a saving throw" is describing what kind of spell you can use Careful Spell on. You cannot use it on spells which do not force a saving throw (for example, firebolt, or mage hand).
If, instead of describing the class of spell, the saving throw clause was describing the timing condition required, we would also have to take it literally when it says "forces other creatures to make a saving throw". That is, if making the save was required at the the time of casting, that save would have to be made by more than one creature; you could not use Careful Spell on a spell that forced a save for a single creature only at the time of spell-casting.
1I am currently working from the SRD and realize that there may be other spells in both categories. However, I doubt there are many more that provoke two saves for the same creature on the same turn the spell is cast.