It would be a sport dominated almost entirely by Europeans until well after WWII, then probably see the Americans show up (the Olympic prestige is certainly greater than Americans' normal level of interest in Grand Prix racing) and then they'd probably have a period of domination before lots of other nations showed up with Grand Prix winners in the years after the early 1960s - Japan, Australia, Brazil, Canada, South Africa, Argentina - and the games got wild.
The big challenge would be whether the games are specifically for drivers, or whether its drivers and constructors - a big question pre-war once the Nazis' massive support of the Mercedes and Auto Union teams resulted in them blowing their competitors into the weeds. If you stick just to drivers, what are their cars? If you use whatever the World Champion car is at the time for the best of the best, that will certainly give an advantage to whatever driver normally races that car - say in 2016, the 2015 World Champion car is the Mercedes-Benz F1 W06 Hybrid, and Mercedes would absolutely love to have its car used in the Olympics (though perhaps less if its technical advances are felt by drivers for other teams in Formula One....), but it gives a major advantage to Lewis Hamilton (presumably one of the United Kingdom's Olympic Auto Racing competitors) and Nico Rosberg (also presumably one of Germany's Olympians).
In addition, what about racers not in F1 at the time, and multiple racers of the same country? 2016 had four Germans - Nico Rosberg, Sebastien Vettel, Nico Hulkenberg and Pascal Wehrlein - competing. The UK has the same problem with Lewis Hamilton, Jenson Button and Joylon Palmer, likewise Finland (Kimi Raikkonen and Valtteri Bottas), Brazil (Felipe Nasr and Felipe Massa), Spain (Fernando Alonso and Carlos Sainz Jr) and Mexico (Sergio Perez and Esteban Gutierrez). In addition, what about nations that didn't have an active F1 driver at the time - Italy, Japan, Argentina, Canada, South Africa and the United States all had long histories of capable Grand Prix racers but none on the grid in 2016, and that's before you point out countries like India and China that would certainly want to be out on the grid.
If you're going for Grand Prix racing in the Olympics, the best bet may be to have a design of Grand Prix car built that is not used in Formula One, and have all of the drivers use that car. The Olympics is about athletic achievement, not engineering ones, so having it be a full team competition gives an unfair advantage to some. That's not all that difficult to find if you're prepared to spend the money to do so - remember that racing would not be a cheap enterprise, and the IOC only allows their sponsors, so the teams cannot have their cars with their sponsor logos on it. But if you could find the money for it, and you have enough of one kind of car to keep it strictly about driver skill, you may have the ability to make this work. It's also possible to have other forms of racing competition like the aforementioned FIA Motorsport Games. Get a major automaker to support the endeavour and you can have them showcase their products as part of the equal cars.
Then you have to decide what the categories are. Grand Prix racing is a given, but what else? Formula E would be ideal for a second open wheel category - electric race cars, lots of people would love that - and then you would need at least a Grand Touring and Touring Car category. A Sports Car category (think Le Mans Prototypes) would be one I'd advocate for, and Karting and Drifting would be great categories as well. I'd also add Rallying (if marathon events can use public roads as venues, why not stage rallies?) for sure, and I'd have a lot of other categories that would be worthy of consideration. Note that if you're gonna go for Olympic Auto Racing, you'd certainly have to have Olympic Motorcycle Racing as well.
Now, for furthering the idea, let's assume Volkswagen-Audi Group is the sponsor - they make ideal cars for a lot of categories, aside from Grand Prix:
Formula E: Audi e-tron FE06
Sports Car: Porsche 919 Hybrid
Grand Touring: Lamborghini Huracan Super Trofeo
Touring Car: Audi RS3 LMS TCR (alternately, you could use the TCR versions of the Volkswagen Golf or SEAT Leon, which are built on the same platform)
Outside of these, you could easily get another sponsor - Ford, Toyota and Hyundai make World Rally Cars and Drifting and Karting would be able to be chosen from a multitude of possible cars.