Imagine a Petunia who loved her sister but was uncomfortable with her chosen world. They would still write to each other and would meet up on occasion before the war got too bad, but otherwise they weren't involved in each other's lives. Petunia is focused on being respectable, making contacts, using those contacts to advance her family's position both personally and professionally, and enjoying the comforts of modern living. She is cunning and ambitious but also loyal to her loved ones and willing to put in the work to better them and herself. Vernon isn't quite as quick-witted as his wife, but he possesses loyalty (for his family, the employees under his management, and his company) in spades, is a hard-worker, and has enough cunning to advance in his company. He often turns to his wife for advice and assistance in making the most of available opportunities.
When Harry is left on their doorstep, they are thrown for a loop and Petunia grieves for her sister, but they never truly consider dumping Harry at an orphanage or harming him. Instead, they try writing to Dumbledore for clarification and to see if a good, magical family could take him in and raise him to understand his place in the magical world. Being the manipulative bad guy he is, he never responds and fully expects and wants them to abuse Harry.
However, Petunia (who, after going through the initial period of grief, goes back over Dumbledore's letter and reads between the lines about how he expects them to treat Harry and his magic) decides to make inroads and contacts in the magical world the same way she does in her world. As she remembers the entrance to Diagon Alley, she starts by going to Gringotts to 1. clear up the issue of guardianship, 2. suggest a yearly or biannual audit of any vaults Harry has or will have access to in order to ensure nothing is stolen (which leads to the Goblins demanding the transcript of Sirius' trial in order to determine Harry's place in the line of the Black heirship), 3. and get the main Goblins on her side working to thwart Dumbledore's machinations. She also asks the Goblins for advice and is advised to 1. request a Ritual of Recall for anything missing from Harry's vaults (*this was in another piece of fanfiction I read), 2. secure a lawyer who can work in both worlds, 3. have Harry get a medical check-up with the Gringotts' healers (thereby discovering the horcrux in his scar and beginning the hunt for the rest), 4. open an account in her own name, and 5. speak with Madame Bones.
During one of her conversations with the Goblins, they suggest she and her family be tested to see if they have any magical affinity (*In one fanfiction I read, both Petunia and Dudley are squibs who can't use wands but do have some magical abilities: Petunia can sway people with her voice and words, Dudley can direct some magic into his fists, and I believe brewing potions was also a possibility). This might be enough to allow Petunia to be named Harry's magical guardian as well.
Petunia and Vernon work together to raise both boys to be "quick to listen, slow to speak, and slow to become angry". They teach them the value of hard work but also the value of delegation when appropriate. They teach them to be kind but not floor mats. They teach the boys to read the people around them and take advantage of any opportunities presented as long as it's honorable to do so. Once Harry is a little older, they explain everything about his parents and Dumbledore's machinations to him. They secure a Muggleborn therapist for Harry to help him work through the mental and emotional trauma as well as help him prepare for re-entering the magical world and dealing with Dumbledore.
Petunia sort of Shanghais Arabella into a friendship (like in Harold the Normal) and makes other contacts in the magical world that help her prepare Harry and the whole family for the coming Hogwarts years (If you are inclined towards a Good Snape, this would be an opportunity to insert him in the story). Petunia's magical contacts also work behind the scenes to help manage expectations about Harry and ensure Harry has some people in the magical world that he knows he can count on. This begins to undermine Dumbledore's power, though he doesn't really realize it until after Harry re-enters the magical world.
Albus (who is the primary bad guy) is complacent, certain that his machinations will bear fruit and supply him with another pawn/weapon, plus Arabella is really vague with her reports (Petunia has gotten her on their side), and the Goblins are working with Petunia and Madame Bones to feed him false information, so he doesn't stick his big nose in their business until it's time for Harry to go to Hogwarts. By then, they are ready for him.