No Nightingales
We, the audience, know what that means. But how do Crowley and Aziraphale know?
Crowley makes it a point to deliver this line in the most 'pay attention, this is important' way possible. "Listen. Do you hear that?" Dramatic pause. And after Aziraphale says he doesn't hear anything, Crowley continues, "That's the point. No Nightingales." Followed by, “You idiot. We could have been… us.”
This is the first time in the show that nightingales are explicitly, openly referred to between Aziraphale and Crowley. Prior to this moment we had God's narration and the song playing on the piano during their romantic celebratory lunch at the Ritz. But no one in-universe has mentioned the song, or alluded to its importance or its meaning to Aziraphale and Crowley.
But during this painful 'breakup' scene, not only does Crowley refer to nightingales, but he does so in a way that is makes it clear that "nightingales" is actually a code between them meaning “us”.
After Crowley says “no nightingales” Aziraphale looks hurt, almost offended. Then Crowley says "we could have been... us" (past tense) and Aziraphale's heart breaks. He quickly looks away because he knows he isn't capable of hiding this pain. We briefly see this about-to-cry expression when Crowley grabs him for the kiss. In other words, Crowley saying "no nightingales" is understood by both of them to mean "our romance, our 'us', has ended before it even truly began."
Think about what this must mean for the rest of their 1941 interaction that we have yet to see. "A Nightingale Sang in Berkeley Square" was published in 1940. I think it’s clear that the first time they heard that song together, they had a breakthrough in their relationship. It might have been a baby step - but it was a significant one. Something happened between them to make "nightingales" code for "us" and it makes perfect sense for that breakthrough to have happened during their romantic 1941 encounter.
But I don't think it was anything as dramatic or earth-shattering as a kiss or a confession. I think it's clear that the bookshop kiss at the end of s2 was their first kiss, and Crowley's "we've spent our existence pretending that we aren't" speech is the first time either one of them as put their feelings for each other in such plain language. Nina's "have you been together long" talk with Crowley, and his subsequent shell-shocked reaction to it, points to Crowley not even really being fully aware that what Aziraphale and Crowley feel for each other looks like romantic love to outsiders because it actually is romantic love.
So what the hell happened in 1941? I am confident that we're going to find out in s3. My guess is that they continued drinking into the night, at some point one of them turned on the radio, and then the song came on. My hope is that Crowley asks Aziraphale to dance, which Aziraphale promptly and nervously turns down (call back to the s2 line "You don't dance.") but this leads them into a discussion about their importance to one another, even if they can't actually demonstrate it explicitly - no dancing, no touching, etc. Then finally, at the end, one of them plainly ties "nightingales" to their relationship in the most achingly romantic way possible. "You'll always have me, until the world ends, until the nightingale stops singing." I'm no writer, but you see what I mean.
I also think "no nightingales" means that when Aziraphale and Crowley finally do reunite and explicitly become an "us" (complete with do-over kiss) we'll get a line of dialogue referencing the nightingale once again.