There are three time machines (or parts thereof) in Terminator Genisys. The one in 2029, the one in 1984 and the one in 2017. Pops and John Connor are able to build the 1984 & 2017 machines because they've had access to the one in the future. But who created the 2029 machine? Are we expected to believe that Skynet, who still has robots driving trucks in 2029, somehow had the creativity to invent time travel? Even if it was capable of such a thing, how would it have known it would need it?
I belive the 2017 machine that we see being invented with the help of John Connor is the time machine that matures for travel in 2029. In Genisys this machine is built at an accelerated rate hence can possibly get completed much ahead in time. But the machine is also damaged in the end so according to the now new timeline they have not mentioned the date of judgement day or the first time travel back.
John from 2029 is sent back with the required knowledge to 2017 to create skynet and the time machine. In part two skynet was shown to be invented because of the T800 arm left behind from part 1. In Genisys they show that the events from part 1 is over written. The first T800 is shown to be quickly destroyed. Because of this skynet will not be created. To avoid that, skynet sends John back to invent skynet. According to the terminator series skynet and john both are born as a result of the predestination paradox. Skynet and John are responsible for their own creation. Skynet sends the T800 which becomes the start point for scientists to start work on skynet. John sends Kyle back and is the one who results in Sarah conceiving John. All of skynet's tech and the resistance leader are part of a closed predestination paradox loop.