Probably a long time, far longer than the litigants would prefer.
The first step is to get the trial court record filed in the appellate court. Texas Rule of Appellate Procedure 35.1 says the record must be filed within 120 days of the judgment being signed. The judgment was signed August 10, 2015, so the record should be file no later than December 8, 2015. The duty to make the filing happen timely is that of the trial court clerk (for what’s filed with the clerk) and the court reporter (for what, if anything in this case, may have been prepared by the court reporter). TRAP 35.3.
After the record is filed comes the appellant’s (the party making the appeal) brief and then the appellee’s (the party defending the decision below) brief. Appellant’s brief is generally due 30 days after the complete record from below is filed. Appellee’s brief is generally due 30 days after the appellee’s brief is filed. TRAP 38.6. Reply brief’s are possible, but the TRAP do not provide any set time for them.
After briefing comes oral argument, unless the parties waive it. TRAP 39. There’s no set deadline in the TRAP for oral argument, however. Scheduling is up to the court and depends on the court’s workload.
After oral argument, the parties must wait for the court to rule. Buying green bananas will not in the least bit be a problem.