There are a few different ways time travel can work if you assume a branching timeline model where each trip back takes you to (or creates) a new timeline. The first, perhaps simplest model is the one that says "when you go back to a given date, you arrive in a timeline that was identical to the one you just came from up until the date of your arrival, after which it can diverge". However, a little thought shows that this model probably doesn't work for T1 or T2 (since this interview with the writers of Genisys seems to imply they do treat T1 as the "original" timeline that is then altered by the appearance of the T-5000, the Skynet-in-human-form who has learned to hop between timelines, while they mostly disregard T3 and take their time travel mechanics from T2). Note that in both T1 and T2, the two time travelers seem to arrive at slightly different moments in the past--in T1 Kyle Reese seems to arrive a little bit after the Terminator, and in T2 the T-1000 seems to arrive a little bit after the T-800. But under the branching-timeline theory above, assuming there was some "original" timeline A where neither member of the pair arrived at that date, it seems as though the second time traveler isn't really arriving in a timeline B that was identical to the timeline A they departed from up until the instant of their arrival. Instead, the movies show them arriving in a timeline B where another time traveler has already appeared a few minutes/hours earlier, already causing a bit of divergence from the history of the original timeline A they came from, where no time travelers appeared on that date. So, these movies don't seem to be consistent with this model of time travel.*
There's a second possible branching-timeline model which involves a sequence of different timelines A, B, C etc., where all the time travelers who depart from one timeline will appear at the appointed hour in the next one in the sequence--so for example all the time travelers who depart from any point in history in timeline A will arrive in timeline B, all the time travelers who depart from timeline B will arrive in timeline C, etc. (A model like this is discussed on this page about a time travel roleplaying game, see especially the scenario towards the bottom of the page involving time travelers who want to interact with Hitler and Marx and the diagram of the different timelines created.) This type of model has some strange consequences. For example, suppose on May 2 2150 of timeline A, I decide to travel back one day to May 1 2150. Then 3000 years later in the 5150 of timeline A, some other time traveler decides to go back 65 million years ago and stop the asteroid that caused the extinction of the dinosaurs. In this case, we will both arrive in the same timeline B, meaning that even though I only intended to travel back one day, I'll find myself in a world where the dinosaurs never went extinct and no humans (aside from fellow time travelers) exist! This second type of model probably wouldn't work very well for science fiction stories since time travelers would almost never end up in anything recognizable as their own past (unless only a few trips back in time are made in the entire history of the universe), but you could make it a little more usable by saying that two time travelers from timeline A can only end up in the same timeline B under certain circumstances, like if only a small time elapsed between the moments they each departed from timeline A, and that otherwise they would end up in different timelines as in the first model.
Some version of this second model can be used to make sense of Genisys (and it can also deal with the problem I mentioned with T1 and T2). Let's say the 2029 we saw at the start of the movie was in timeline A, where no terminators arrived in 1973, Kyle Reese's experiences in 1984 played out as they did in T1, Judgment Day happened in 1997, and in general everything happened the same way as in the T1 timeline until the moment the T-5000 appeared. Then in this timeline A, sometime after the T-5000's appearance--probably after it had converted John Connor--someone (probably the T-5000 itself, or possibly the Skynet who was "native" to timeline A) decides to send a T-1000 back to 1973 to kill Sarah Connor, and someone else sends "Pops" the Terminator to 1973 to protect her. Also, the converted John Connor (the T-3000) is at some point sent back to the 2010s to work with Cyberdyne.
Then in this second model, all these time travelers sent from timeline A will appear at the appointed time in timeline B, with the ones arriving later finding a timeline that's already been altered from what they remember by the time travelers that arrived earlier. In particular, when Kyle Reese arrives in the 1984 of timeline B, he finds himself in a timeline where Sarah Connor has been accompanied by "Pops" since 1973, which wasn't true in the original timeline A. Likewise, when the John Connor T-3000 talks to Sarah, he still remembers her raising him in the 80s and 90s in timeline A (as shown for example in the scene where Sarah goes to the bunker with Kyle and says she used to go there with her dad, then the T-3000 appears and says something like "Mom, did you honestly think I wouldn't remember the place you spent so much time with Grandpa?"), even though the Sarah of timeline B skips forward to 2017 without having any children, and even if she does have one she probably won't take him to that same bunker after what the T-3000 said.
*It would actually be possible to reconcile T1 and T2 with the first branching timeline model, but only by assuming multiple other timelines were created besides the ones we saw onscreen. Say for example that in some "original" timeline A, the T-800 terminator and Kyle Reese were sent to 1984, but no T-1000 or T-800 were sent to 1995. Then in timeline B, Skynet sends back a T-1000 in an additional attempt to get rid of John Connor, and the Resistance sends back the T-800. Under the first model, this would create two new timelines, call them C and D--in C only the T-1000 would arrive, and in D only the T-800 would arrive. In timeline C we can assume young Connor is killed and Skynet wins, thus the Skynet of timeline C has no further need to send anyone more back. But in timeline D, when 2029 rolls around again, the Skynet of this timeline may again decide to send back a T-1000, creating yet another timeline E. And since it was already a part of the past of timeline D that a T-800 appeared in 1995, it's consistent with this model for the T-1000 to arrive in a version of 1995 where a T-800 has already been there for a few minutes or hours.