Slightly Ambiguous, But Previously Cast Spells Probably Still Work
The crux of the matter here is that the word "cast" typically is the same in the past and present tense. As such, the precise wording of globe of invulnerability stopping spells "cast from outside the barrier" could intend to refer to past, present, both, or exclusively one (for example, only spells cast before the globe existed), and it would have exactly the same grammatical structure.
We have guidance, however, in the other sentences of the spell's description. We are told (PHB, p. 245, bold added):
Such a spell can target creatures and objects within the barrier, but the spell has no effect on them.
This sentence uses terms exclusively in the present tense. It doesn't say that the spell can "have targeted" creatures (indicating that they targeted them in the past and are still targeting them), but rather speaks of only spells that are now attempting to target a creature once the globe of invulnerability is in effect. As such, it is likely that both the intended timing of the targeting and being affected are the same: meaning the globe of invulnerability blocks spells against something inside the globe when the other spells are cast.
The other interpretation leads to madness
The grammatical ambiguity permits people to interpret this spell as they wish. But keep in mind that many complex cases could arise if globe of invulnerability blocked previously cast spells. For example, let's say a wizard casts mirror image on themselves, then moves 20 feet and (in the next round) casts globe of invulnerability. Was the spell mirror image cast from outside the globe or not? Such a question is needlessly open to interpretation (can you be said to be "outside" the globe when it doesn't exist? Since the globe appears around you, is the source of mirror image you, or your previous location?), but would not even come up if the globe only blocks subsequent spells.