The Moon is 384,400 km or 1.28 light seconds from Earth. That means that it takes light, or any electromagnetic communication like radio, 2.6 seconds to make it from the Earth to the Moon and back again. Actual communication delays are even longer, due to having to relay the signal from the receiving antenna (sometimes halfway around the world) to its destination via radio or undersea cable.
That time delay WAS there. If you look at the timestamped transcripts of Apollo 11 communications, you can verify for yourself than no response was received less than 2.6 seconds after the query. (Additional delays are there, of course, due to other factors - the astronaut pausing to take a reading, flip a switch, think for a moment, etc.).
However, in many archived media recordings, the lightspeed delays have been edited out. In the specific case you mention, the call from Nixon in the White House to the astronauts, there is only one place in which Nixon finishes speaking and the astronauts respond. There’s a delay between them. After the astronauts respond, Nixon replies once more to end the call, and of course there is no delay there. Nixon heard the astronauts at the same time we all did, even though they had transmitted 1.26 seconds earlier.
The recording has an additional interesting feature. Prior to Nixon coming on line, and afterwards, there are transmissions from Houston to the astronauts, and their responses. In these transmissions, you can hear the echo of Houston’s words, caught by the astronauts’ open microphones. And those echoed words illustrate the lightspeed delay quite nicely. You see, if there is no delay and you feed back a signal through an open mic, you get a feedback howl. But these repeated signals were delayed enough that they did NOT cause feedback.