It's going on and has been since the announcement late December. The main issue is that supplies of batteries are low, publicity is extremely high, and there are a lot of people who are looking for a new battery because they see it's a discount for an OEM battery. The original announcement was that replacements at the promotional price would begin late January, to give Apple time to order and receive the substantial number of batteries they would need from their suppliers. Before this time, many balked at the official $79 price (in the United States) and might even go for an aftermarket battery replacement even though it might affect whether Apple is willing to service the phone in the future.
My understanding of how it works is that once you bring it in they will run the standard battery diagnostic. In the past the basic rule was that Apple or Apple Authorized Service wouldn't perform a battery replacement unless the battery registered below the industry standard 80% battery health of a new battery, or otherwise had failed. But under this promotion they will for $29, but just once unless it fails the battery diagnostic. You get one chance at a $29 battery replacement when your battery is registering as "healthy". During the promotion, any time the battery is fails the diagnostic, they'll replace it for $29. But once you've gotten that $29 offer on an otherwise healthy battery, you've used up that one ticket.