Only the part that is contingent on the attack roll itself is doubled
There seems to be intent that it is not doubled, as evidenced by the Green Flame Blade question below:
In a tweet Crawford specifically answers a question about the subsequent damage on Green Flame Blade.
If you critically hit with the attack part of Green Flame Blade, and
are greater than level 5, do you roll extra dmg against the 2nd
Answer from Crawford:
The splash damage of green-flame blade isn't affected by the attack
critting. Think of the attack as process X & the splash damage as Y
Coupled with Sage Advice 2020 p13:
Can spell attacks score critical hits?
A spell attack can definitely score a critical hit. The rule on
critical hits applies to attack rolls of any sort.
I do realize that JC's Tweets are no longer dogma but until definitive clarifications come from Sage Advice they will have to suffice. That said JC's Tweet on Witchbolt criticals dovetail nicely with Melf's Acid Arrow in so far as the adjudication of damage on subsequent turns.
What if the Witch bolt crited? Since there's no attack roll for the
subsequent damage, would it roll only the first as a crit or all of
them as crits? (I assume only the first one, but eh)
Witch bolt's initial attack can, indeed, crit, but the damage
delivered via actions on subsequent turns is unaffected by the crit. A
crit isn't intended to affect the damage of other
These lead me to believe that the only part of a spell attack that is doubled is the part that is dealt as part of the actual attack roll. As the subsequent round's damage would be the 'Y' or even 'Z' in this instance.
I went through the podcasts and couldn't find anything else, mostly the spells section dealt with targeting in the one from Jan 17, 2017.