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On the other hand, it seems reasonable that the effect of Heightened Spell is explicitly being restricted to just the first saving throw provoked by sorcerer spells that can lead to multiple saves over multiples turns such as 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.

On the other hand, it seems reasonable that the effect of Heightened Spell is explicitly being restricted to just the first saving throw provoked by sorcerer spells that can lead to multiple saves over multiples turns such as 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, Incendiary Cloud, and Power Word Stun.

'When you cast a spell' refers to the timing of the use of Careful Spell, but not the forced save

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 actually says (emphasis mine):

Why specifically call out its first saving throw, when you could simply say, "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 that 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.

¹_{I 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.}

'When you cast a spell' is unclear, but we can refer to Heightened Spell for context

Multiple possible meanings

It is not immediately obvious what 'when you cast a spell' means here, because the phrase is used to mean different things at different places in the rules.

Sometimes 'when you cast a spell' sets up things that can only happen at the time of spell casting, like choosing who is permitted to enter and leave a Tiny Hut. If this was the case here, then since no saves are made for Web at the time of its casting, Careful Spell could not be used to protect allies.

But sometimes 'when you cast a spell' sets up things that happen for the entire duration of a spell, like when you upcast a Wall of Fire, and the additional damage applies over every round the spell is in effect. If this was the case here, then Careful Spell could protect your allies from any saving throw imposed by Web for the duration of the spell.

Unfortunately, since the phrase itself is used in different ways, there is no way to know for certain which way it is intended to be used in Careful Spell based on a reading of the text of Careful Spell itself. Either interpretation produces a result that seems reasonable, as the varying answers to this question attest.

However, we can compare Careful Spell to other metamagic features that use the same phrasing and do not have as ambiguous an interpretation.

Comparison to Heightened Spell

Heightened Spell says (emphases mine):

If, in the context of metamagic features, 'when you cast a spell' was meant to apply to effects that last for just the casting of the spell itself, there would be no need to specifically call out its effect as being on the first saving throw only, since it would already be understood that only a save made at the time of spell casting would apply.

On the other hand, it seems reasonable that the effect of Heightened Spell is explicitly being restricted to just the first saving throw provoked by sorcerer spells that can lead to multiple saves over multiples turns such as 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.

If Heightened Spell contains language that explicitly restricts its use to the first save and Careful Spell does not, then it stands to reason that Careful Spell is meant to apply to any and all saves provoked by the spell it is used with, so long as the phrase 'when you cast a spell' is used consistently between the two metamagic features. And while 'when you cast a spell' does mean different things in different contexts, I can think of no good reason why it would mean different things in two examples of the same context, that is, two metamagic features that affect creature saves against the spells they are used with.

It seems to me that the most logical inference is that Careful Spell affects all saves. Thus, one may protect an ally from Web by using Careful Spell, even when the save is not made on the same turn the spell is cast, and even if multiple saves are made over multiple rounds.

Heightened Spell says (emphasis mine):

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

Heightened Spell actually says (emphasis mine):

Why specifically call out its first saving throw, when you could simply say, "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 that saving throw."