Not only did the allied powers not try to erase Volkswagen, they basically created the company we now know. The following comes from James Flink's The Automobile Age.

As was mentioned elsewhere, the Volkswagen Beetle had its origins in the KdF-Wagen, part of the Third Reich's efforts to make Germany into a mass automobile culture. The idea of a Volksauto certainly predated the Nazis - it had been used by a number of German manufacturers to describe their affordable (and often rear-engined, streamlined) offerings, going back to the 1920s; and whether the KdF-Wagen was the brainchild of Ferdinand Porsche (as the narrative typically goes), shamelessly stolen from Czech automaker Tatra, or shamelessly stolen from German-Jewish auto journalist and engineer Joseph Ganz, the story of it's development from that point is not much in doubt.

The development of a state-sponsored volksauto crystalized into the KdF-Wagen in August 1938, when it was announced that the Volkswagen project would be sponsored through the KdF (Strength through Joy) division of the German Labor Front (DAF). The DAF announced a plan by which "a worker could purchase a Volkswagen on a 5-mark weekly layaway plan for 990 RMK, including insurance garage, and maintenance costs" (quoting directly from Flink). It's worth noting that at the time of this announcement, Porsche and the German government had only produced prototypes for the car; they did not even have a factory capable of meeting the production needs yet. Nevertheless, Germans flocked to the KdF-wagen program. Exchanging their Marks for stamps to fill out a savings book, 336,668 Germans invested 280 million RMK into the program; the little KdF-wagen prototype was celebrated with parade floats and reams of Nazi propaganda. The final terms of the plan were a far cry from its lofty announcement, however: rather than the advertised 990 RMK price tag, the total cost under the savings plan came out to 1190 RMK; and even worse, the language of the KdF contract "did not in fact obligate the state to even deliver a car once the final payment had been made" (again quoting from Flink). In fact, fiscal mismanagement and ballooning construction costs at the new plant in Lower Saxony meant that production had not even begun by the time of the Nazi invasion of Poland and the suspension of the program.

In what Flink describes as "the world's biggest investment swindle," over three hundred thousand Germans paid monthly installments for cars that were never built; instead, the manufacturing plant built with their savings turned out war materiel using slave labor until its capture by the Soviets. Despite the cancellation of production, the Nazi government continued to collect payments through the end of the war. upon capture, the Soviets confiscated the remaining funds from the savings accounts. (As an aside, the newly-formed West German government spent decades in litigation with KdF-wagen savers, eventually settling for a payment of either 600DM in credit towards a new VW Beetle or 100DM in cash)

So, by war's end there was no Volkswagen to speak of - and it's hard to say that one had ever existed in the first place. It would have been perfectly simple for the occupation government (Lower Saxony falling in the British occupation sphere) to simply sell off the remaining assets and level the complex. Two-thirds of the plant had already been destroyed by allied bombing, and the slave-laborers smashed the remaining machinery and ransacked the offices and barracks after their SS guards (who I would imagine did their own share of records-destruction) fled. Time magazine reported that "every telephone had been torn from the walls, every typewriter had been sledgehammered to junk, every file and record had been scattered and burned."

Nevertheless, the British almost immediately began the process of converting the factory back to civilian auto production. They renamed the town from "Stadt des KdF-Wagens bei Fallersleben" (Strength-through-Joy Car City) to Wolfsburg, and enlisted the few remaining German workers in cleaning debris and putting machinery back in working order. Initially, the factory continued to build military "Kubelwagens" (German jeeps) for the British occupation forces, but by October 1946 ten thousand civilian Volkswagens had been built and the factory workforce had exploded from 450 in April 1945 to nearly 8,000 in August 1946. The British authorities running the operation sent a production car back to England along with a proposal to include the Wolfsburg factory in the war reparations, but a 1946 commission lead by William Rootes (head of the Rootes auto manufacturing conglomerate) "predicted that [the company] would collapse within two years and that even if the Germans did get the car into production" (from Flink) and that "the vehicle does not meet the fundamental technical requirements of a motor car [...] as regards to performance and design it is quite unattractive to the average motorcar buyer, it is too ugly and too noisy [...] to build the car commercially would be a completely uneconomic enterprise."

After Ford Motor Company rejected an offer for the factory, the British Military Government installed former Opel executive Heinz Nordoff as general manager, who oversaw the name change to "Volkswagenwerk" and the transfer of control from the British to the FRG and then, after a lengthy dispute for ownership between the FRG and the state of Lower Saxony, VW was made a public company in 1960.

As an aside, the British government's instrumental role in creating VW did not necessarily belie an immediate enthusiasm for the car among the British public (if Sir Rootes' report gives any indication). According to Flink, the British press regularly referred to the beetle as "Hitler's Car," and some 75 percent of beetles arriving in the UK were damaged by vandals.

I hope this helps!

Modified from an earlier answer of mine

The conventional date for the takeoff of the FRG's long-term export success is 1951/52 when the Korean War caused a disruption in international trade that allowed German firms to establish themselves as a major player in international commerce. From the Korean War to reunification, FRG export growth as a whole remained relatively steady, even during global cyclical downturns like the oil shocks of the 1970s.

The postwar success of West German exports did alarm a number of commentators in Western Europe where memories of German occupation and bombing ran long, but there was no equivalent panic or hand-wringing in the United States. FRG products remained somewhat invisible inside the US market for a number of reasons. Firstly, the FRG's primary trading partners tended to be Western European and the North American market remained peripheral to FRG trade. Moreover, FRG manufacturing tended to focus on capital goods like machinery, steel, chemicals, and medium-technology items like precision instruments. Although the FRG did produce high-end consumer products such as Sennheiser's audio equipment, these were often dwarfed by more mundane industrial products. As such, the FRG's penetration of the US market tended to be in items not purchased by an average consumer, such as Krupp-Thyssen elevators and escalators or x-ray machinery.

The one notable exception to this trend was automobiles. Volkswagen had begun significant exports into the US by the mid-1950s and its famous "think-small" campaign helped to cement it as an icon of the 1960s. Similarly, both Mercedes and BMW also penetrated the US market. Yet prior to the 1970s, FRG cars tended to fill in market niches that Detroit either ignored or felt beneath their effort. For example, the VW Beetle was one of the cars of choice for both women and households needing a second car, demographics that Detroit expected to be filled by the used car market. Mercedes and BMW became associated with an alternative luxury market, one which both Detroit and US consumers already associated with European-made products. As such, the wider cultural reaction to these German products on US shores did not resurrect the ghosts of the Second World War since German cars initially were not displacing US manufacturing, but instead serving a need US industry did not attempt to fill. A side bar in Melvin J Lasky's rather fawning 2 October 1955 NYT's article" The Volkswagen: A success story," captured this complacent mentality. After noting to its readers that although the Beetle was the fastest-selling foreign car in the US, the sidebar contended:

But despite the popularity of this small German car, there is little likelihood, in the opinion of the automotive industry, that Detroit will soon follow suit. The manufacturers are convinced that the demand is not big enough for small cars. They point out that a year-old car- fully equipped - is cheaper than a new Volkswagen, and they add that it costs almost as much to tool up to build a small car as it does for a large one. In the event of public demand, however, the industry is prepared to update plans it had long shelved for small cars.

Similarly, a 10 April 1957 Wall Street Journal article on European economy cars quoted both Harlow Curtice of GM and L. L. Colbert of Chrysler somewhat curt dismissal of these smaller automobiles, with GM's executive noting they were uniformly "substandard" and Colbert noting Chrysler had designs for small cars dating from before the war, but had no plans to start production of them.

This somewhat complacent attitude to this West German product was in strong contrast to the British reaction to the arrival of the VW Beetle in which tripped off alarms about the ailing British car industry and other forms of industrial malaise. Advertisers for British cars like the Sunbeam Imp and MG 1100 explicitly tried undercut Beetle sales both domestically and abroad by comparing themselves favorably to the German product. The Imp was specifically priced $100 below the Beetle. Pressure from British manufacturers led higher tariffs on imports in 1965, to which VW responded by imposing a ban on British parts. French car maker Renault likewise saw Volkswagen as its major competitor and its advertisements both at home and abroad stressed the superiority of French cars to German ones.

This anti-Axis sentiment would reach US shores in the 1970s, especially with the advent of Japanese compact cars and consumer electronics whose sales outstripped their American counterparts at a time of increased economic uncertainty. Instead, the American reaction to FRG's economic resurgence in the 1950s and 60s was a sort of smug self-vindication that unlike the Soviet-backed GDR, "America's" Germans had learned the valuable lessons of industrial capitalism and were now respectable members of the global economic community- for an example of this mentality- see the Warner Brothers cartoon Word of Mouse in which a German mouse is taught the benefits of mass consumption and production by his more worldly American cousin. Lasky called VW's Wolfsburg Dearborn-on-the-Luneburger-Heide and noted the city "is 'American' only in the sense that it too is a great melting pot, with a rootless population to find opportunity and pursue happiness." Lasky also drew a picture of its chief Heinrich Nordhoff that likened him to a figure that melded old-world charm and the savvy practical modernism American businessman. Vinton McVicker's 20 Mar 1956Wall Street Journal article "People's Car: Defying Detroit Dogma, Volkswagen Triumphs In Global Markets" favorably compared both VW and Nordhoff to Henry Ford. A 10 October 1962 Wall Street Journal piece on Nordhoff explicitly stated:

Dr. Nordhoff wasn't a Nazi. He was and is a patriotic German. The Hitler government needed experts, and Dr. Nordhoff qualified. By 1942, he was director general of the Opel-werke in Brandenburg, then the largest truck factory in Europe.

The NYT's 1968 obituary for Nordhoff also took pains to point out he had never joined the NSDAP and treated his initial blacklisting by the occupation governments as an unfortunate interlude in a career of West German industrial development. Although both the firm and its chief's Third Reich past was an open secret, at one point in the war, forced laborers made up half of the Opel-werke in Brandenburg's labor force, the American press preferred to tell of a more edifying story of West Germany prospering under American tutelage.

Perhaps ironically, the stigma of the Third Reich upon German manufacturers actually grew as direct memories of the Second World War faded. German industrialists like Alfried Krupp conducted an intensive public relations campaign in the immediate aftermath of the war portraying themselves as reluctant accomplices of the NSDAP who were forced to compromise their private enterprises by a criminal regime. By an large, these efforts were successful and American plans for tribunals against these industrialists fell apart. These early public relations efforts muddied the waters over the actual degree of responsibility of these firms for war crimes and cultivated an image of a German industrialist as a patriotic anticommunist. The American occupation authorities conceived of responsible and strong industry as a firewall against Communist infiltration of the German workforce. Despite notable efforts like William Manchester's The Arms of Krupp, very few commentators in the West took industrial firms to task for their collusion with the Third Reich. The GDR's constant harping about an continuities between the FRG elite and the Third Reich also led to writing off such criticism as repetition of the Eastern blocs' wider talking points of the Cold War. The wider historical reevaluation of Germans' coordination and cooperation with the NSDAP regime in the 1970s and 1980s helped to uncover the broad connections between the state and German industrial firms, many of whom actively competed for allocations of slave labor. Lawsuits against German firms in the 1980s and 1990s by survivors of forced labor helped strip away the vestiges of the postwar whitewashing of these firms' responsibilities. This efforts collectively led to a greater public awareness of the connection of German firms to the Third Reich so much so that it has almost become axiomatic that whenever a German company is caught doing something wrong, such as the VW emissions scandal, there is no shortage of commentary that will mention the unsavory association of these firms with the Third Reich.

Sources

Brady, John, Beverly Crawford, and Sarah Elise Wiliarty. The Postwar Transformation of Germany Democracy, Prosperity, and Nationhood. Ann Arbor: University of Michigan Press, 1999.

Logemann, Jan L. Trams or Tailfins? Public and Private Prosperity in Postwar West Germany and the United States. Chicago: University of Chicago Press, 2013.

Owen Smith, E. The German Economy. London: Routledge, 1994.

Rieger, Bernhard. The People's Car: A Global History of the Volkswagen Beetle. Cambridge, Mass: Harvard University Press, 2013.