Green Flame Blade and Booming Blade

[Dr. Cliché]

Both Green Flame Blade and Booming Blade require you to make melee attacks as part of their casting.

Do these attacks use your melee stat (Str/Dex) or your casting stat?

[DivisibleByZero]

It's a melee attack, so it uses your melee attack modifier, and therefore your melee attack stat.

[Rivuzu]

Sort of.

When you cast Booming Blade, you attack with a melee weapon. Treat that as a normal melee weapon attack, including the damage. Booming Blade and GFB are additional effects that happen on hit.

Not compatible with multi-attack characters though, as the attack itself is considered Casting a Spell and not the Attack action.

[DarkKnightJin]

Sort of.

When you cast Booming Blade, you attack with a melee weapon. Treat that as a normal melee weapon attack, including the damage. Booming Blade and GFB are additional effects that happen on hit.

Not compatible with multi-attack characters though, as the attack itself is considered Casting a Spell and not the Attack action.

This. You're making a normal melee weapon attack as part of casting the spell.
EK with War Magic gets to cast BB or GFB and make a BA attack.
Otherwise a character with extra attack would have to choose between casting the cantrip or attacking twice. Or more, if they're non-EK Fighters.