Imagine a Petunia who saw Lily, the favored child, grow up into a manipulative bully who even turned her back on the best friend she had because she valued shallow opinions and popularity more than the boy who taught her so much. They would occasionally write to each other (Lily to brag and Petunia to maintain a connection to her last blood relative), but 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, but they never truly consider dumping Harry at an orphanage or harming him. Instead, they try writing to the Potters first, then 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 if the Potters were so set on getting rid of him. Being the manipulative bad guy he is, Dumbledore never responds and fully expects and wants them to abuse Harry while the Potters don't respond because they don't care. They are too busy playing up and basking in the fame the other twin (wrongly labeled as the Boy-Who-Lived) brings them.
However, Petunia (who 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. arrange for the proper management of Harry's vaults (this is when she discovers that Harry is the oldest twin and the heir for the Potter Lordship and tertiary heir for the Black Lordship after Sirius [here you decide what role you want Sirius to play in this story]), 3. and get the main Goblins on her side working to thwart Dumbledore's and the Potters' 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 a vault in their 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. When Harry is a little older, they secure a Muggleborn therapist for him and explain just what his parents chose to do as well as what they've been able to discern about Dumbledore's machinations. Harry sees the therapist to work through the mental and emotional damage the truth causes and prepare for the inevitable future meeting with his parents and 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 tell trustworthy wixen a little about Harry and how the Potters tossed him aside, thereby undermining the Potters' and Dumbledore's support, though they don't fully realize it until Harry is properly brought back into 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 and the Potters.