Apparently it's intended to scale, though that's not totally obvious
As you point out, in the original printing of the PHB, the second bullet point of the Martial Adept feat read (PHB, p. 168):
If you already have superiority dice, you gain one more; otherwise, you have one superiority die, which is a d6.
This clearly indicates that Martial Adept's superiority die does scale up with the Battle Master fighter's Improved Combat Superiority feature, turning into a d10 at 10th level and a d12 at 18th level.
However, the very first errata in 2015 reworded that part of the Martial Adept feat to read as it does now (emphasis mine):
You gain one superiority die, which is a d6 (this die is added to any superiority dice you have from another source).
However, that errata document (and every other one up through 2017 or so) summarized the change as follows:
The superiority die is added to any others you have, no matter when you gain them.
Some people (including me!) were initially confused when the 2018 PHB errata quoted the new text, thinking it was a new change distinct from the previous printings (though it's not marked as "[NEW]" in the errata document). However, the only new thing in that regard is that the 2018 errata consistently includes direct quotes of the new/current wording, rather than paraphrasing it as previous errata PDFs did (which can potentially lead to confusion, as in this case).
Many people understood this to mean that the Martial Adept die does scale with the Battle Master's superiority dice regardless of when they get the feat vs. the subclass. However, the revised wording of the feat could also be interpreted to mean exactly the opposite - that this superiority die is separate from and in addition to any existing superiority dice (i.e. that it's a flat d6, and does not scale with the Battle Master's superiority dice).
Eventually, lead rules designer Jeremy Crawford unofficially clarified how Martial Adept interacts with Battle Master fighters in an April 2020 tweet:
If you're a Battle Master fighter and take the Martial Adept feat, the superiority die from the feat improves when you gain the Improved Combat Superiority feature, which intentionally makes no distinction between a die you get from the class and a die you get from the feat.
In order to understand Crawford's ruling, we then have to look at the wording of the Battle Master fighter's Improved Combat Superiority feature (PHB, p. 74):
At 10th level, your superiority dice turn into d10s. At 18th level, they turn into d12s.
Given the parenthetical in the revised wording of the Martial Adept feat, it's unclear to me whether the superiority die from Martial Adept remains a d6 from Battle Master fighter levels 3-9, or if it becomes a d8 as soon as you have any Battle Master fighter levels. (That may be worth asking as a separate question.)
However, what is clear from Crawford's interpretation is that when you gain the Improved Combat Superiority feature, it is intended to extend the benefit to Martial Adept's additional superiority die as well. Since the feature merely says "your superiority dice" (rather than "your superiority dice from this archetype" or "your superiority dice from the Combat Superiority feature"), it does apply to all your superiority dice, no matter where they're from.1
If the intent was for Martial Adept's superiority die to scale, the initial 2015 errata's change in the wording of the feat seems unnecessary/self-defeating, since the first errata clearly indicated that it scaled (since it just added to your existing superiority dice if you had them already).
The reason for the errata may have been that the original wording might suggest that Martial Adept's superiority die would scale if you had Battle Master fighter levels first before taking the feat, but not if you took the feat and then gained levels in Battle Master fighter (as you wonder in the question). In contrast, the revised wording merely says "this die is added to any superiority dice you have from another source", with no mention of the Battle Master fighter.
However, if the change was merely out of concern about the rules around the Battle Master needing to account for other sources of superiority dice, then the designers could have instead removed the relevant wording from Martial Adept entirely and explicitly clarified in the Battle Master's own features that Martial Adept's superiority dice scale with it. Instead, Martial Adept's clear but flawed wording was replaced with vague/unclear wording (that has resulted in confusion and rules arguments for years now).
Especially given the confusion around this issue, this question would be a good one to provide an official ruling on in the Sage Advice Compendium, rather than relying solely on Crawford's tweet (which is not an official ruling) to understand the interaction.
1 The same logic extends to the interaction of the Battle Master fighter's Improved Combat Superiority feature with other sources of superiority dice in playtest content. The only instance of this in a published book is in Tasha's Cauldron of Everything (p. 41), which adds the Superior Technique option to the fighter's Fighting Style feature, letting any fighter gain a d6 superiority die and 1 maneuver (rather than 2 maneuvers, as Martial Adept grants).
The playtest versions of the Cavalier and Scout fighter archetypes in UA: Kits of Old, as well as the Monster Hunter fighter archetype from UA: Gothic Heroes, also have superiority dice and an identical Improved Combat Superiority feature, so they'd interact with other sources of superiority dice in the same way.
