Ed Teller's H-bomb test (Ivy Mike) was a success, but his bomb was so large that the DoD called for a million+ pound thrust rocket engine in order to get it over the Pole and to the destinations. Von Braun knew two things -- that he was likely the only one who could build one, and that it was his chance to send humans to the Moon.
A second team had a successful H-bomb design which didn't have the yield of the Teller bomb, but with an H-bomb that's not really a problem.
By the time the DoD changed its mind about its needs, VB's F1 engine was well into development. Components had been tested by '56 and the cavitation problem was solved by '59. Development had been taken over by NACA, the forerunner of NASA.
It was ready and waiting when JFK got ticked off by Soviet crowing about its successes in space.
Ike's failure to allow VB's earlier staged rocket to orbit a satellite (Ike said it would be "provocative") a year before Sputnik was an embarrassment. The safety conscious decision to orbit a chimp instead of an astronaut in the first Mercury, a few weeks before Gagarin's April '61 flight, was another embarrassment. Eventually these failures ticked off JFK. He called NASA, asked if the US could land on the Moon first, they said yes, and he stated the goal to Congress in May 1962, and gave the more famous "we choose to go to the Moon" that September.
“Our Germans are better than their Germans.”