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> Babai’s new result says that solving graph isomorphism takes slightly longer than polynomial time – not quite placing it in P, but significantly shifting the needle for the first time.

What is the needle and how did it shift significantly for the first time?

Edit: They already knew it was in NP, but now they know it is not in P (or not quite in P)?

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If it was proven that it's not in P then we would have shown that P!=NP which would be HUGE.

No, this is "merely" finding a more efficient algorithm for a problem which we still don't know exactly where belongs. From a complexity theoretical viewpoint nothing is changed. It's still possible that graph isomorphism is in P.

Now if graph isomorphism is shown to be NP-complete then this result would imply that every problem in NP can be solved in sub-exponential time which would have huge practical consequences.

All of P is in NP. The result claims that it is close to P, when before it was probably expected to be much harder than P.

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