As used in the series, the word "obviously" is a tip-off that something is false or at least non-obvious in some way.
It's used mainly in Episode 1, first when Crowley says they just sent him up to make some trouble, not specifically about the apple. Aziraphale says "Obviously. You're a demon, it's what you do." It seems like a simple statement, but there were long discussions historically about the concept of the "Fortunate Fall" where eating the apple was vitally necessary to the Plan. Making that particular trouble turns out to have been the right thing to do.
Aziraphale next uses it ironically, when Gabriel claims that he never sullies his celestial body with gross matter while standing there wearing a cashmere suit. "Obviously. Nice suit."
He's unconsciously ironic when he tells Crowley that "Obviously" Heaven will win. It's nowhere near obvious; at best it's a 50-50 chance.
Gabriel tells Aziraphale that his efforts to influence the Antichrist are obviously doomed to failure. He is both ironically right and completely wrong. Aziraphale is working on the wrong kid (though I head-canon that Warlock was improved by the attempt), but Adam could be influenced. He was influenced toward the light by his parents and friends, and that was strong enough to let him turn away from his origin.
When Crowley notices that Adam has named Dog, Aziraphale is doubtful. Crowley asks, "Would I lie to you?" and Aziraphale repeats the first usage identically: "Obviously. You're a demon, it's what you do." After 6,000 years and eleven years of working together closely, he still doesn't entirely trust Crowley, yet Crowley isn't lying. I don't recall offhand a single instance when Crowley ever lied to Aziraphale.
It's used again in Episode 3 when they argue about holy water in 1862, and Crowley says "I don't need you." Aziraphale says "The feeling is mutual. Obviously." Crowley repeats "Obviously." They're both wrong, but they don't, apparently, talk about it again for 105 years, despite having made up in 1941.
[ETA: OOPS, I missed one in Episode 3, Hard Times:
1601:
Az: You are up to no good.
Cr: Obviously.
This one is harder to shove into my theory. Crowley is there to try to get Aziraphale to work the Arrangement again, and succeeds. There is one odd bit: at that moment, Crowley is to Aziraphale's right, instead of left. That usually signifies that Crowley is actually more in the right than Aziraphale is. Aziraphale does succumb to the temptation to avoid work, and loses the coin-toss to boot. But he also makes puppy-eyes at Crowley, who volunteers to help Shakespeare, so Crowley actually is eventually "up to good" (assuming you like the play).]
Another character uses the word. (Two if you count when in Episode 4 of the Script, Michael uses it about "there are no back channels", but that use doesn't appear in the Series as shown.)
In Episode 5, the Quartermaster says that returning to Earth without a body is obviously ridiculous, because angels can't possess humans and Aziraphale has no body. We learn later that it's because without physical eyes, Aziraphale can't see where he is. Still, returning is not entirely ridiculous, because angels can possess humans. They're just taught that they can't.
ETA again: When Aziraphale is studying Agnes's book, Crowley calls asking if there is any new information. Aziraphale says there isn't any, if there were he would tell Crowley, "obviously." Well, he hasn't quite worked it out completely yet, but very shortly after that he makes the telephone call that confirms where Adam is. He seems to have understood the part about Tadfield being where the Antichrist still is.