It already has, as of the signing of Chupp's final order. The first deadline is the record from the trial court has to be lodged with the appeals court within 120 days.
I'm not sure if that's happened yet, but at least some of it has been filed.
After that, the appellant (the party appealing) has 30 days to file their opening brief. This is the operative document in the appeal. The appellee (the party being appealed against) has 30 days from then to file their reply brief. I would expect in this case there will be at least one round of sur-replies after this. These supplemental briefs generally get shorter.
Then the appeals court either sets the case for oral argument, generally a brief hearing less than an hour which may be entirely consumed by questions from the judges on the panel, or if the court doesn't feel it is necessary and it isn't required by some other rule, it can simply decide the case.
After the oral argument, the appeals court issues an opinion basically whenever it feels like it.
I had assumed previously this would be on an accelerated schedule and had posted the rules for that in a previous post, but the papers actually filed have a checkbox for whether it's accelerated and this isn't checked, so apparently it isn't. At least I don't see any indication of that.
The rules for the TCPA make certain appeals, like interlocutory appeals and appeals of denials, automatically accelerated, but this apparently doesn't apply to an appeal after a final judgment fully granting a TCPA.
But I'd anticipate, just based on previous cases, that they'll handle this relatively quickly, so it will be more on the order of seven months than over a year. Texas appeals courts seem to grasp that TCPA cases should be heard somewhat expeditiously.