There can be argued by that its possible but also that it is not:
Possible:
Does the spell effect only one? Yes it does which mean that no doub't the twinned spell works and it should work.
Is the melee attack a magic attack, well its used in the cantrip action so it cant be regular attack. Which indicate it should be possible.
now here is why it can be argued as
semi not possible:
Not possible:
The spell itself is possible we need to determine the mechanism:
Boomingblade as part of the action make a melee attack against one creature, else the spell fails.
Twinned spell can be cast if the spell you use only target one creature
Now boomingblade directly state that as a part of the action make a melee attack. Well, in order to by the text to cast the twinned version you no longer need to make an additional melee attack in order to succeed casting the spell. Since the melee attack states it directly requires an action, you cant do that as two actions even though you twin it. The spell will be twinned but it will be the effect of the boomingblade so to speak the magic.
Now people would argue that the attack itself is magic and an additional magical sword appears or something like that, but since boomingblade directly states that the melee attack that is required is part of an action you need two actions in order to do two melee attack on two creatures...but the twin effect still apply meaning that after you hit one time with the melee attack the magic itselfis twinned and you both the target you hit and another target within range will be sheathed in booming energy.
Now to investigate even further we need to know about magic...
Often confusion arises when the Antimagic Field spell is used, an Armor of Invulnerability is worn, or the Beholder's anti-magic ray is employed, whether a given character's ability will still func...
rpg.stackexchange.com
The sage advice compendium made a little relation to when is it magic or not...so if it should be possible to twin cast the melee attack itself then a new problem comes into light, cause that would mean according to the checklist that the melee attack thereby would be magic. And suddently we have a even stronger boomingblade than before if that is the case. Cause if you let the twin work it would be a spell and thereby per default magic attack, which it says in its text it is not by the wordings: "target suffers the attack's normal effects " which indicate that the melee part is not magical. Therefore conclusion is that if you twin cast boomingblade you do:
One melee attack
Two booming sheath attacks and effect.
Now noone can actually say one is more correct than the other since the text can be argued and read on two different terms where noone can actually say one is more right than the other one...
We must therefore wait till final confirmation from Jeremy E Crawford. All Jeremy has confirmed as of now is that it is possible to twin cast the spell...question that remains is if its possible to do two melee attacks or not...