> the war effort
...which definitely included motorized vehicles to carry troops, cannons, etc. It's the same world as ours with the same technology in existence. Winston and Julia did ride a train to the country, if I recall correctly, at one point.
Cars were, as you say, both recreational and utilitarian in 1948, and yes, of course they existed, I am aware of the Model T and so forth. What they were not, though, was culturally central.
Driving was still (aside from professional use, as I said) a recreational activity practiced by the affluent - look at Mr. Toad in The Wind In The Willows, for instance, a rich dilletante given to manias who has discovered motorcars, to his near financial ruination, at the start of the novel. They were personal thrill rides, more like owning a speedboat, or else a means to some end, like feeding one's family by trading in heavy goods. A middle class family living in London would buy a car so they could drive to the country every weekend, but not to practically get Mr. Londonman to work on time.
That's 1908, but I would argue that the place cars occupied in the public imagination forty years later in 1948 would still look a lot more like that, with the car (for those able to access one at all) a central family asset like the radio or the dining table, rather than an individual's personal accoutrement that served to signal their attitude and affluence.
Not that that paradigm didn't exist at all - hot rod culture had already started also, as had biker culture. There were already people who had taken on automobilia as their raison d'etre all over the world, without a doubt. But they were outliers: fringe groups, 1%ers, speed-obsessed engineers and dangerous (to themselves, mostly) amateurs. If we were talking about computers, things were at the Homebrew Computer Club stage of things.
A poor person working a job classified as "unskilled" in North America today is just as likely to own a car as a computer programmer, by necessity rather than inclination, which is what makes the absence of them in 1984 feel strange to a modern reader like OP. This is a thing that my urbanist friends get very hot under the collar when we start talking about it, and it's a cultural shift/ripoff that took effect long after Orwell wrote 1984.
I think his non-mentioning of cars in 1984 merely illuminates the mindset and world from which Orwell was writing, rather than representing a conscious choice to say to the reader, "these poor wretches don't even have their own motorcars!"