[RPG] Does this specific build to use Booming Blade at a range of 20 feet work

The wording of "Long-Limbed" as you state is clear (emphasis added):

When you make a melee attack on your turn

Let us have a closer look at opportunity attacks (PHB p73, emphasis added):

You can make an opportunity attack when a hostile creature that you can see moves out of your reach. To make the opportunity attack, you use your reaction to make one melee attack against the provoking creature. The attack occurs right before the creature leaves your reach.

...

You also don’t provoke an opportunity attack when you teleport or when someone or something moves you without using your movement, action, or reaction.

The polearm mastery feat extends the OA condition (emphasis added):

While you are wielding a glaive, halberd, pike, or quarterstaff, other creatures provoke an opportunity attack from you when they enter your reach.

Three things are important here: 1. What are the conditions for OA and 2. when are they executed.

1. OA depend on whether an enemy leaves or enters (due to Polearm Master) your reach.

2. The attack is executed during the movement phase of the enemy, so clearly during their turn and turns are not overlapping.

3. Forced movement like shoving does not trigger OA, even it the enemy is leaving or entering your reach. Forced movement by dominate or similar does indeed trigger OA, as that uses their movement.

Condensed answer: Your reach during an opportunity attack is 10 feet due to the reach property of your weapon. Your OA can be executed when an enemy is -on its own- about to leave or enter but not move within your reach. During your own turn, your reach for a melee attack is 15 feet.

There is some confliction in your question but I'll assume you want to make Melee Weapon Attacks at reach.

The highest reach would be 35 feet with a Giant-sized Bugbear Battlemaster Fighter/Way of the Four Elements Monk's unarmed strike whose flesh has been warped by an ichor-coated weapon. It costs 1 Superiority Die and 1 Ki point.

• Base reach is 5 feet
• Bugbear gives +5 feet
• a Potion of Giant Size gives +5 feet
• Lunging Strike (Fighter: Battlemaster) gives +5 feet
• Fangs of the Fire Snake (Monk: Way of the Four Elements) gives +10 feet
• In Avernus, weapons coated in demon ichor have the chance to warp flesh:

The target’s arms become tentacles with fingers on the ends, increasing its reach by 5 feet.

This combo works because of the Player's Handbook Errata, which changed future printings of the Player's Handbook. It reveals how the unarmed strikes can be used for the battlemaster's lunging strike.

Instead of using a weapon to make a melee weapon attack, you can make an unarmed strike ... You are proficient with your unarmed strikes.