So as long as you qualify for Sneak Attack, it applies. In your example, the melee attack would trigger Sneak Attack.

You can even get sneak attack on an opportunity attack later in the round because it's only limited to once per turn. Sneak attack is not an attack action, it's a triggered event based on its prerequisites.

The official Sage Advice Compendium confirms this:

Can you use green-flame blade and booming blade with Extra Attack, opportunity attacks, Sneak Attack, and other weapon attack options?

Introduced in the Sword Coast Adventurer’s Guide, the green-flame blade and booming blade spells pose a number of questions, because they each do something unusual: require you to make a melee attack with a weapon as part of the spell’s casting. First, each of these spells involves a normal melee weapon attack, not a spell attack, so you use whatever ability modifier you normally use with the weapon. (A spell tells you if it includes a spell attack, and neither of these spells do.) For example, if you use a longsword with green-flame blade, you use your Strength modifier for the weapon’s attack and damage rolls.

Second, neither green-flame blade nor booming blade works with Extra Attack or any other feature that requires the Attack action. Like other spells, these cantrips require the Cast a Spell action, not the Attack action, and they can’t be used to make an opportunity attack, unless a special feature allows you to do so.

Third, these weapon attacks work with Sneak Attack if they fulfill the normal requirements for that feature. For example, if you have the Sneak Attack feature and cast greenflame blade with a finesse weapon, you can deal Sneak Attack damage to the target of the weapon attack if you have advantage on the attack roll and hit.

Rules designer Jeremy Crawford unofficially clarified via Twitter that it refers to using your movement, as in "I move X feet" - not a generic "move", as in "I scratch my nose":

When booming blade refers to moving, it means movement in the game's normal sense: moving X feet.

Moreover, standing from prone, while it uses movement, does not count as a move for the booming blade spell, as Crawford has also clarified:

Standing up costs movement but moves you nowhere. To move while prone, you crawl or use magic (PH, 191).