For the gist of the problem you should first read this message from Apple. https://www.apple.com/iphone-battery-and-performance. They have now changed the commencement date and have already started the replacement service.
I would advise you to book with the genius bar of your local store as there are bound to be many others with the same need. In addition to that it has become known that stock is low and many countries are waiting supplies. Because of the enormity of this operation, people should show patience.
It also appears that the old battery may not need to be tested to qualify for the discounted rate. The time needed to replace the battery should not exceed an hour. The offer will be worldwide and available through Apple retail stores, Authorised re-sellers and Apple service centres. It is likely to end in December 2018.