Without von Braun's influence in the West, would they even have tried the race at all? And if not, would Korolev (and the others) have had a better chance to unfold things more as they planned and with less pressure for "spectaculars" to upstage the West?
Whatever the fate of ze Germans - the US and Soviet rocket scientists were vastly different. America, too, had talented rocket scientists (Karel Bossart, M. Atlas Balloon Tanks) but they were scattered across the aerospace industry. Crucially they had no direct access to the President - it was a) their top boss > the industrial lobby > the military / NACA / NASA > the President.
In the Soviet Union the Yangels, Korolev and Chelomei had direct access to Stalin or Mister K. later on.
America had rocket scientists aplenty without a real need for von Braun. Viking, Redstone, Thor, Atlas, Titan, Navaho were done without the german team. Von Braun main usefulness was, well, he was
different - an excellent, popular speaker through the Collier series. He tried shaking america inertia for rockets before Sputnik yet met mixed success.
The main reason why America resisted rocketry before Sputnik (and even with von Braun) relates to its military. Without NASA they were the main funding for space - NACA and the scientists were nothing by comparison. Yet the US military had bombers aplenty - nearly ten thousands of B-29s, B-50s, B-36s, B-47s, B-52s. Plus the Navy carriers and their attack planes. Intercontiental rockets ? for what ?
The Soviets had no such luxury, no aircraft carriers, no warm water ports to dock them, and their Tu-4 force was pathetic. Further atempts by Tupolev (Tu-80, 85, 95) were obsoletes before flying. Myasischchev (3M, M4) was a little more successful but then supersonic bombers raised the level of difficulty and expense again (M-50). In the end USSR hedge all its bet on rockets, with tremendous success. Although the missile gap was an idiocy and the Nedelin disaster a major setback, it took until the Saturn I (1961) before america bet the R-7 (1957).