My own experience suggests that if the applicant is getting impatient then it's best to log into the DPS website to view the status online and not wait for some written notice of an application deficiency that may never arrive.
I applied for a renewal slightly less than six months prior to the license expiration date, then promptly forgot about it. Their website warned applicants that new finger prints might be required. I joked to a friend that they probably routinely request new prints to buy time since they were so busy.
I remembered it after a few months and decided to follow up through their online status application. Sure enough, they wanted new prints. Furthermore, a line in the status page said I had been notified. Nope. I got no notice other than my own eyeballs on their webpage. BTW, that was 135 days after I mailed the application.
Furthermore, the criminal background check apparently requires the prints, because they didn't begin that check until they had the prints in their system.
In any event, the license arrived in the mail nine days after the new prints were taken at Identogo, so kudos to the DPS for their prompt attention to the matter once they had everything in hand.
In their defense, record numbers of Texans have applied for a handgun license which is a good thing. But they were swamped. Maybe their workload has lightened a bit now that the gun salesman of the decade is no longer president.
As for statutory deadlines, they have 60 days for a new application and 45 days for a renewal once the materials have been received. Reference: §411.176. REVIEW OF APPLICATION MATERIALS and §411.185. LICENSE RENEWAL PROCEDURE found online here.
Meanwhile, if you are waiting on an original or renewal and get impatient, go log on at The License to Carry a Handgun service to check the status. If your last prints were on paper you'll probably have to get them retaken electronically. If your online status report says you need to submit new prints, don't waste time waiting for written notification, just follow the instructions at LTC Fingerprint and Photo Information and take care of it. The fee for prints at identogo.com was $10.21 with tax. Cash was not accepted at the outpost where I went.