Following the Rules as Written, no, the spells do not work together. However, Crawford has said removing this interaction is not what they intended when reworking Booming Blade and Green-Flame Blade. He tweeted on November 13, 2020:

This change has nothing to do with prohibiting or allowing Shadow Blade to combine with Booming/Green-Flame Blade. It's about fixing those two cantrips. As DM, I'd allow those them to combo, since I make liberal use of the rule on improvised weapons.

So since Crawford allows it, I think most DM's should. The reason for the change was that casters were having swords magically appear from their Component Pouches. A Component Pouch contains every spellcasting component without a cost. Previously, Booming Blade did not have a cost, but did require a weapon as the component. Thus, following Rules as Written, every time you cast the spell you could pull out a weapon from your component pouch. They had to fix this, thus the 1sp component was added, since the component now had a cost it couldn't be pulled out of the component pouch.

I hope this helps in making the case to your DM that the two can and should work together!

Following the Rules as Written, no, the spells do not work together. 

However, Crawford has said removing this interaction is not what they intended when reworking Booming Blade and Green-Flame Blade. He tweeted on November 13, 2020:

This change has nothing to do with prohibiting or allowing Shadow Blade to combine with Booming/Green-Flame Blade. It's about fixing those two cantrips. As DM, I'd allow those them to combo, since I make liberal use of the rule on improvised weapons.

So since Crawford allows it, I think most DM's should. The reason for the change was that casters were having swords magically appear from their Component Pouches. A Component Pouch contains every spellcasting component without a cost. Previously, Booming Blade did not have a cost, but did require a weapon as the component. Thus, following Rules as Written, every time you cast the spell you could pull out a weapon from your component pouch. They had to fix this, thus the 1sp component was added, since the component now had a cost it couldn't be pulled out of the component pouch.

I hope this helps in making the case to your DM that the two can and should work together!

Following the Rules as Written, no, the spells do not work together. However, Crawford has said removing this interaction is not what they intended when reworking Booming Blade and Green-Flame Blade. He tweeted on november 13, 2020: "This change has nothing to do with prohibiting or allowing Shadow Blade to combine with Booming/Green-Flame Blade. It's about fixing those two cantrips. As DM, I'd allow those them to combo, since I make liberal use of the rule on improvised weapons."

So since Crawford allows it, I think most DM's should. The reason for the change was that casters were having swords magically appear from their Component Pouches. A Component Pouch contains every spellcasting component without a cost. Previously, Booming Blade did not have a cost, but did require a weapon as the component. Thus, following Rules as Written, every time you cast the spell you could pull out a weapon from your component pouch. They had to fix this, thus the 1sp component was added, since the component now had a cost it couldn't be pulled out of the component pouch.

I hope this helps in making the case to your DM that the two can and should work together!

Following the Rules as Written, no, the spells do not work together. However, Crawford has said removing this interaction is not what they intended when reworking Booming Blade and Green-Flame Blade. He tweeted on November 13, 2020:

This change has nothing to do with prohibiting or allowing Shadow Blade to combine with Booming/Green-Flame Blade. It's about fixing those two cantrips. As DM, I'd allow those them to combo, since I make liberal use of the rule on improvised weapons.

So since Crawford allows it, I think most DM's should. The reason for the change was that casters were having swords magically appear from their Component Pouches. A Component Pouch contains every spellcasting component without a cost. Previously, Booming Blade did not have a cost, but did require a weapon as the component. Thus, following Rules as Written, every time you cast the spell you could pull out a weapon from your component pouch. They had to fix this, thus the 1sp component was added, since the component now had a cost it couldn't be pulled out of the component pouch.

I hope this helps in making the case to your DM that the two can and should work together!