Sounds like you're looking for the Continuing Time series by Daniel Keyes Moran. It's no longer a trilogy, and was always meant to go for several books, but due to lack of popularity and sales, Moran has had trouble with publishers and may have self-published the fourth book (it's not immediately clear - it looks like it's only available digitally). According to Moran's biography, the long wait for the fourth book was due to his focusing on raising his children.
This series does not fit all of your criteria, but I think it has the right vibe. There is a large group of children with psychic and telepathic abilities. They do grow up over the course of the books. The world is definitely cyber-punk, and the second book focuses a lot on the digital/online world as its main character is a "web dancer" (the in-universe term for a black-hat hacker).
However, unless this happens in the fourth book (which I haven't read) no one from the Castenaveras family (which make up the main characters of the books) becomes a leader in any government.
If this is the series you are looking for, then you might be conflating it with the Ender's Game series, by Orson Scott Card (also a three-name author). The children in Ender's Game are the products of a kind of eugenics system, they also grow up through the series, they are not psychic but they are very intelligent, and at the end of the first book,
Ender's brother and sister have manipulated the global population using online arguments and Ender's brother (Peter) has become the Hegemon - like the president of the world.
The publication dates are slightly off from what you're looking for. Both the first books of The Continuing Time and Ender's Game series are from the late 80s.
This synopsis of the first book is from memory, since my copies are in storage:
The setting is the future, New York City (ca. 2030 - 2064). Most of the world is unified under one government that is the evolution of the United Nations - I think because of a small-scale nuclear war that scared most people into wanting unity over liberty. Unfortunately, the UN has become a bit draconian. One example is that driving your own car is mostly outlawed, generally everyone is required to have their cars automatically controlled by a traffic control system. Because of overpopulation there is a Ministry of Population Control that forces most young women to be sterilized. Also many crimes have extreme punishments. Some who have violated the car control laws have been executed. The enforcement and military arm of the United Nations government is the UN Peace Keeping Force, or PKF for short.
A secret experimental program of the UN has created a group of genetically engineered enhanced humans through gene splicing and cloning.
Actually, we learn during the story that a mysterious figure that can travel through time and space made a change to one of the experiments that caused the spliced genes to successfully replicate into a human who has psychic and telekinetic abilities. None of the other experiments before were successful, and it takes several years before the program is able to actually genetically engineer another human from scratch.
The first successful engineered human was named Carl Castenaveras. Some time later Carl is partly cloned (they sort of created a fraternal twin) named Jany McConnell. Carl and his "sister" are psychic and telekinetic. Eventually the scientists are able to custom assemble genes to create more telepaths (they copy the important gene complex in Carl and Jany). Also Carl and Jany have natural born twins named David and Denice. All of the telepaths live in a compound in Manhattan with Carl and Jany acting as parents, trying to raise the children to be effective, responsible telepaths and members of society. Because they are all the result of a government experiment, their status as humans and citizens is uncertain, so Carl spends a lot of time fighting for full legal status for his family with a lawyer friend.
At the end of the first book, distrust of the telepaths grows. The telepaths are successful in their legal battles and are able to act independently of the PKF, which the UN government finds unacceptable. Protesters, some of whom are hired by the PKF to stir up trouble, are constantly gathered outside the compound.
The situation comes to a head and the government kidnaps David and Denice. Carl leaves the compound to rescue them. The PKF tries to negotiate with the telepaths and one of the PKF gets trigger happy and there's a bloodbath - only the PKF officer in charge survives the encounter. The telepaths all join together and pool their power to defend themselves and create a large, hugely powerful group mind that causes insanity many nearby residents of Manhattan. In response, the PKF nukes the compound. Only four from the family survive - Carl, David, Denise, and the young boy Trent, who is the only one who is not a telepath because there is a mistake in one of his gene complexes. Trent was staying with a friend of the family because he couldn't handle being around telepaths without being one. Carl frees David and Denise but dies in the process. Trent is actually arrested but uses his hacker skills and the help of an independent AI to escape custody.
Any synopsis of the second or third books would be almost entirely spoiler text, so I don't see the point. They are available pretty cheaply as Kindle books. The second one, The Long Run, is one of my favorite SF novels, so I would say at least reading the first two is worth it.