So if the local Skynet is doing it's job, it's presumably going to start watching out for Time Travelers. Now that I'm thinking about it, since Skynet learns at "an exponential rate" it's probably going to be constantly inventing new and better weapons to do it's job and protect the Earth (In between rounds of World of Warcraft). Since the local Skynet wouldn't be hampered by blowing up most of the world's infrastructure pointlessly, it would have a higher starting techbase and resources than OTL Skynet and would begin making all sorts of stuff as contingencies for everything it can think of.
This could lead to a hilarious parody world where periodically superweapons suddenly pop out of nowhere and unleash unthinkable firepower at something, apparently at random, and nobody has any clue why until Skynet posts a tweet about it. People can't even do anything because Skynet, worried that it didn't have enough resources, played the stock market and bought itself it's own land and servers that it protects without government intervention, after all an incompetent leader could prevent it from doing it's job.
So now in this brave new world suddenly a mile-long submarine nobody knew about surfaces and unleashes a swarm of missiles that does incredible localized damage in Belize and it turns out, time travelers had come to destroy us all. A week after that an orbital laser precisely vaporizes a single person who is revealed to have been the carrier for a horrific incurable bioweapon, Skynet also already took care of the creators of the disease. Then a month later the moon splits open to reveal Skynet has built a superlaser inside and it begins opening fire into deep space, after a year there's the most brilliant meteor shower in history, and we find out an alien civilization had fired RKKVs at our solar system and Skynet took care of it.
After a few decades Skynet's Twitter feed looks like something out of One Punch Man from the sheer number of ridiculous threats it's taken out before they could get started.
Something like "dear diary,alien try to invade Eart.I beat them except cute catgirls which was send to Japan.
It was 12.375 invasion in my career".