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The Alchemist artificer's 5th-level Alchemical Savant feature states (WGtE, p. 180; E:RftLW, p. 58):

Whenever you cast a spell using your alchemist’s supplies as the spellcasting focus, you gain a bonus to one roll of the spell. That roll must restore hit points or be a damage roll that deals acid, fire, necrotic, or poison damage, and the bonus equals your Intelligence modifier (minimum of +1).

Green-flame blade (SCAG, p. 143) is not on the Alchemist's spell list, but it lists "a weapon" as the material component for the spell. This is presumably the weapon used in the attack.

If an Alchemist casts green-flame blade using their alchemical tools (e.g., a beaker) as an improvised weapon attack, would the fire damage benefit from Alchemical Savant?

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Not unless the improvised weapon was also used as the focus

While foci replace the use material components in spells, material components are nonetheless not foci per se. Alchemist's supplies used as an improvised weapon may function fine as the material component for Green Flame Blade, although it's doubtful, but that doesn't make them the spellcasting focus for the spell. That said, if you can use them as a spellcasting focus for the spell (e.g. if your group's resolution to this underspecification-related issue ends with Green Flame Blade counting as an alchemist spell for you for some reason, or, more likely, because your arcane focus for whatever is giving you the cantrip is also an artisan's tool of some kind), you can use them instead of the weapon material component, and that would let you make use of the class feature. You are also incorrect that the material component and the weapon with which you strike must be the same thing; you could use a tiny wooden spear in your spell component pouch as the component and then attack with a dagger you have in your other hand as the attack that's part of the spell.

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