There is no reason to think that the time limitations prescribed by TRAP 21.4 and 21.8(a) do not apply to a motion based on art. 40.001. The enactment of 40.001 in 1993 was merely a legislative change to former Rule 30(b)(6)(changing "new" to "material") and an effort to make clear that the reasoning in such cases as Alford and Meriwether was not acceptable. Professor Dix has referred to "the 30-day limitation on filing motions for new trials on the ground of newly discovered evidence." Texas Practice:Criminal Practice & Procedure, sec. 50:26 (2011). I am thinking the CCA will act pretty quickly on a writ application on this ground, if the applicant can meet the requisite standards.
Newell correctly points out the jurisdictional hurdle. See Drew, 743 S.W.2d at 222.
That chapter used to contain a statutory time limit for filing a motion for new trial, but those provisions were repealed to give the CCA the rulemaking authority to set deadlines. The deadline for filing the motion was basically then ported over to Rule 21.4. Admittedly, Article 40.001 stayed in the Code whilst Article 40.05 disappeared. But as Martin notes in his citation to Drew, Article 40.001 was governed originally by the same deadline that currently exists in the Rules of Appellate Procedure. So the legislative change wasn't a broadening of the trial court's authority to rule on the motion for new trial and the continued existence of Article 40.001 in the Code doesn't translate into an independent vehicle for the trial court to grant a new trial unfettered by deadlines where none had existed before.
But that's just me.
State v. Moore, 225 S.W.3d 556 (Tex. Crim. App. 2007) provides an interesting history discussion on the shift of motions for new trial to the Rules of Appellate Procedure, albeit in the context of an amendment to a timely filed motion for new trial. If you're feeling masochistic.