Husband prepares car with essentials for nuclear meltdown in local facility in USA and need for fast getaway. Neighbour borrows wheel jack. Husband has spare so he isn't left without one. Family rush off when meltdown starts. Wife fiddles around delaying the need to leave urgently and wants to divert to get relatives but there is no time to lose. They only just squeeze onto congested freeway and escape the city. At petrol station he requires attendant to spill some on ground so he can check it's not water. He leaves empty-headed wife and son with all their money at station and heads out to remote USA with only level-headed family member, the daughter. Can anyone recall the story name or writer?
"Lot", a novelette by Ward Moore; first published in The Magazine of Fantasy and Science Fiction, May 1953, available at the Internet Archive. A sequel, "Lot's Daughter", appeared in The Magazine of Fantasy and Science Fiction, October 1954, also available at the Internet Archive; the two stories were published together in book form as Lot and Lot's Daughter.
It's not a "meltdown at local facility", it's nuclear war. Everything else matches.
In the wake of a nuclear attack, a man puts into effect his well-organized plan to evacuate his family, and comes to realize that not all of them are likely to help with the task, or even not to impede it.
Mr. Jimmon was conscious of Warbinn's glowering at him and resolutely refused to turn his head. He pretended not to hear him yell, "Only wanted to tell you you forgot to pick up your bumper-jack. It's in front of our garage."
Mr. Jimmon’s stomach felt empty. What if he had a flat now? Ruined, condemned. He knew a burning hate for Warbinn — incompetent borrower, bad neighbor, thoughtless, shiftless, criminal. He owed it to himself to leap from the station wagon and seize Warbinn by the throat. . . .
"What did he say, David? What is Mr. Warbinn saying?"
Then he remembered it was the jack from the Buick; the station wagon’s was safely packed where he could get at it easily. Naturally he would never have started out on a trip like this without checking so essential an item. "Nothing," he said, "nothing at all."
[. . . .]
It was 4 when they got to Santa Barbara and Mr. Jimmon faced concerted though unorganized rebellion. Wendell was screaming with stiffness and boredom; Jir remarked casually to no one in particular that Santa Barbara was the place they were going to beat the bottleneck oh yeh; Molly said, Stop at the first clean-looking gas station. Even Erika added, "Yes, Dad, you'll really have to stop."
Mr. Jimmon was appalled. With every second priceless and hordes of panic-stricken refugees pressing behind, they would rob him of all the precious gains he'd made by skill, daring, judgment. Stupidity and shortsightedness. Unbelievable. For their own silly comfort—good lord, did they think they had a monopoly on bodily weaknesses? He was cramped as they and wanted to go as badly. Time and space which could never be made up. Let them lose this half hour and it was quite likely they'd never get out of Santa Barbara.
[. . . .]
Furiously Mr. Jimmon determined to preserve the civilization in Erika. [. . .] protect her night and day from the refugees who would be roaming the hills south of Monterey. The rifle ammunition, properly used—and he would see that no one but himself used it—would last years. After it was gone—presuming fragments and pieces of a suicidal world hadn't pulled itself miraculously together to offer a place to return to—there were the two hunting bows whose steel-tipped shafts could stop a man as easily as a deer or mountain lion. He remembered debating long, at the time he had been preparing for It, how many bows to order, measuring their weight and bulk against the other precious freight and deciding at last that two was the satisfactory minimum. It must have been in his subconscious mind all along that of the whole family Erika was the only other person who could be trusted with a bow.
[. . . .]
"Tell you what I'll do. I'll pay for each gallon as you pump it. In advance." He drew out a handful of bills; the bulk of his money was in his wallet, but he'd put the small bills in his pockets. He handed over a five. "Spill the first one on the ground or in a can if you've got one."
Why should I give him ideas? As if he hadn't got them already. "Just call me eccentric," he said. "I don't want the first gallon from the pump. Why should you care? It's just five dollars more profit."
[. . . .]
Mr. Jimmon beckoned his wife around the other side of the wagon, out of sight. Swiftly but casually he extracted the contents of his wallet. The 200 dollar bills made a fat lump. "Put this in your bag," he said. "Tell you why later. Meantime why don't you try and get Pearl and Dan on the phone? See if they're OK?"
[. . . .]
"All right," said Mr. Jimmon. "Get in Erika."
Some of the light shone directly on her face. Again he noted how mature and self-assured she looked. Erika would survive—and not as a savage either. The man started to wipe the windshield. "Oh, Jir," he said casually, "run in and see if your mother is getting her connection. Tell her we'll wait."
"Aw furcrysay, I don't see why I always—"
"And ask her to buy a couple of boxes of candybars if they've got them. Wendell, go with Jir, would you?"
He slid in behind the wheel and closed the door gently. The motor started with hardly a sound. As he put his foot on the clutch and shifted into low he thought Erika turned to him with a startled look. As the station wagon moved forward, he was sure of it.
"It's all right, Erika," said Mr. Jimmon, "I'll explain later."
He'd have lots of time to do it.