I've uploaded a sketch of an equidistant projection that goes out to 51 degrees, which is to say 3060 nautical miles or 5680 kilometers, from New York City at the center. Judge for yourselves which possible route is least bad! The westernmost point of the Azores as shown by this shoreline trace is about 35 degrees from the target. Note also that a straight line from it to the target (any line from the center to any point represents a great circle route in this projection) approaches over open waters, whereas a line from the tip of Brittany passes very near Cornwall, and one from Norway would thread between the Orkney and Faroe islands, barely missing northern Scotland, squeak past southern Iceland, and then approach NYC on a long track overland in North America, crossing over Nova Scotia and all of New England! For either of the more distant origin flights to work well, they'd have to dodge, adding more miles to the trip, and Norway IMHO is clean out; veering one way to avoid one British/American base just brings them closer to another.
Like I said, for a one-time stunt, it would be possible for the Azores to be unoccupied by anybody (though OTL the British moved in on them pretty quickly) and for the Luftwaffe to move through them, in and out; it would require some covert preparation. And give the Portuguese regime some semblance of plausible deniability, but that won't wash in the context of the full-on war; Salazar would be blamed even if he honestly had no idea the Germans were going to violate his neutrality that way, for letting it happen. And he can't just declare war on the Germans; they'd come pouring in through Spain (and force Franco to declare for the Axis too) and take his government apart in short order. Too bad the Allies couldn't move into Portugal on his invitation, but surely Salazar doesn't want his country to become a battle front and it is very doubtful the Allies have forces ready and able to move in fast enough to stop the Wehrmacht and Luftwaffe. So Portugal was neutral, perhaps, but it can't be afterward. Salazar doesn't have to think the Axis will win; he has damn little choice if the Germans decide to use his country.
Franco is somewhat another matter. OTL he temporized, stalled, and weaseled, because unless the Axis was poised to crush Britain in very short order, his country would be doomed to severe hardship if the British could blockade it. If Hitler pig-headedly decided to simply march over Spain, I don't suppose the Spanish forces (decimated quite recently by the civil war, though of course then built up--with Axis help) could do much to slow him down. But they could do something, Axis forces being compelled to come in through restricted passages over the mountains or around them on the coasts. If he could temporarily delay them he might buy time for an Anglo-American landing he supports. I've speculated about this before, but I am far from being able to war-game it enough to guess how long loyal Spanish forces could hold off the Germans nor how fast could the Allies put how much into Spain. Odds are, the Germans get past the mountains and Spain becomes once again a battlefield, on a scale that puts the devastation of the Civil War into the shade. Nor is this a great scenario for the Allies--it puts them on the European mainland all right, but they already have that in Italy, and they wouldn't be a lot better off than in Italy, having to contest the terrain with stubborn German resistance, with the Germans being able to fall back to the mountains at their back just as in Italy. The Pyrenees are not quite the Alps but still, it would be a hard slog, just to get into southern France, and then another long hard slog across France to Germany. If Franco were very friendly, he might anticipate being betrayed by Hitler sooner or later, and seek to covertly prepare to secretly introduce hidden, disguised Allied units to reinforce the northern border, to hold the Germans there and buy time for a full on muster of an invasion force behind the line, and bring them up. This would save Iberia from the devastation of war (except for Luftwaffe bombing, but in such a scenario British and American air forces would move in quickly and parry the Luftwaffe, then go on a bombing offensive of their own). But it is a harebrained scheme; I don't suppose sufficient Allied units could be brought in with enough secrecy to seriously delay the German advance around or over the mountains.
Anyway Franco was not sympathetic to the Allies. Recognizing the British held the cards of vital overseas trade and held the overseas colonies (his original power base having after all been in Morocco) hostage, he did not want to antagonize them, but neither did he wish the Soviet Ally anything good, and cursed the Western Allies for joining with them. Insofar as Free French forces were a factor, he wouldn't like or trust them either, France having aided the Republican side of the Civil War, and any conservative French military men who might have preferred to see the aid go to Franco would be sitting in Vichy France on the other side; the Free French would be filtered to be the most anti-Fascist. His best and only option was neutrality, and it was within Hitler's power to violate it any time he decided the effort would be worthwhile. Surely if Hitler chose to do that, I suppose Franco would try to make the best of it by openly throwing in with him, and hoping (absurdly of course) that with Spain in the Axis as an independent partner rather than a puppet state, that the Axis might yet prevail.
He could hardly stop the air strike against the USA even if he wanted to; I suppose the Spanish air forces might have delayed it and sounded an alarm but then Hitler would just vengefully roll over Spain and it would become a pathetic puppet state if not incorporated into the Reich on similar terms to Poland or Bohemia. Best case for himself then would be calling in the Allies belatedly, which would be pretty terrible and might not even work--therefore the Allies might not even try, resolving to simply blockade the peninsula and reinforce the defense of Gibraltar. The air strike probably then forces Franco to take the dire consequences of open war with the Allies and try to get what benefit he can from openly joining the Axis. In which case Portugal is dead meat too.
OTOH, the particular Amerika-bomber mentioned in comments was indeed high-maintenance, heavy on wing loading, presumably heavy on the runway too, so it would require a reinforced runway, concrete or anyway prepared with steel matting, and a long one because it had high take-off speed. To get the range and bomb load required any plane would have to be heavy; the only way to make it STOL would involve some combination of even bigger wings (which would make it slow and vulnerable to interception) or even more powerful engines. The designs developed OTL after all did not satisfy the leadership as being adequate, yet they were at the extreme limit of what German aviation could accomplish, so a much better design seems unlikely.
Yeah, it is a bit of a paradox, which is why the USA never did suffer any airstrikes OTL of course! If the Germans could have prepared heavy runways in the Azores (and in Iberia somewhere, though in a pinch I suppose these long-range planes might have flown from France to the Azores nonstop) it would be doable, but how could they accomplish this secretly? The British surely have spies there, and as a neutral country the Portuguese possessions would be hard-pressed to keep big secrets. I suggested maybe an ostensibly civil project, to prepare long hard runways for future transAtlantic air liners, but that would probably have been too alarming for the Anglo-American alliance to let slide and would trigger an occupation.
So--either these Amerika-bombers are a different design, that can use relatively primitive runways, perhaps using jet-assisted takeoff or some such. Or they have to stage a different way. Even if both Franco and Salazar had declared for the Axis openly, say in the last days of the Battle of France along with Italy, this would merely have guaranteed the loss of all their overseas colonies immediately, especially the Azores. Hitler had plenty of troops to garrison them--but neither the sea power nor the air power to place them after a Portuguese DOW, while trying to pre-position them would violate Portugal's claims to neutrality and again expose the islands to a quick British takeover--this one with no apologies to Lisbon. One might fantasize a combined air-lift/cargo U-boat reinforcement of the Azores, but I'm pretty sure the RN would have been up to blocking it, and taking control of the islands handily behind the screen of their fleet. Once taken, suitable airfields would surely be built--and firmly in Allied hands, to bomb Iberia and points east rather than distant America to the west.
Given the difficulty of a suitable design to take advantage of Azores airports, perhaps it would be better to instead stage out of France--out of Brittany, which is closer to America than Norway is, in particular. To deliver a big bomb load, this might require airplanes even more grandiose than the OTL designs.
It is a matter then of which is easier to design, a plane that can strike at 6000+ kilometers range, or one that can land and take off on poor primitive airstrips in the Azores. Although I like the idea of staging through the Azores, it is a one-time stunt since the Allies can move in on these islands and at any rate deny them to the Luftwaffe if not so easily seize them for themselves. A plane that can make it from Norway can do better from western France.
Since the Germans can ill afford to send wave after wave to America anyway, I suppose Hitler might recognize that this is a one-time trick. Perhaps the easiest way to stretch the OTL designs to be both attainable and practical on existing airfields would be to design in jet-assisted takeoff, on a grand scale, suitable to allow an unimproved Azorian field to launch a laden plane. It could come in to this airfield almost empty, thus being lighter and allowing a short landing run, then get loaded up including a battery of hypergolic liquid rocket engines (in frequent use in the German forces). This would be risky as all hell, but say Hitler enjoys his Devil's luck once again this time and just the once, the engines all work as planned, the fuel is successfully transported (in submarines!) without blowing up and/or poisoning the sub crew, and the Americans get no warning of the strike until the planes are practically upon their targets.
I don't know what degree of coastal air defense the USA had in wartime, but I would not be surprised to find it amazingly primitive and minimal; the major defense of the USA is sheer distance and if we look at it cold-bloodedly, it is more rational to deploy advanced radar and heavy defenses forward against the Axis than defensively along US borders, since the damage Axis raiders could do at such extreme ranges from all safe bases for them would be minimal. I daresay this strike does compel the US forces to beef up the coastal defenses, but they would probably do it on the cheap and largely for show, to reserve more resources for where it really counts; still the mission is a partial success for the Germans insofar as it does divert resources from the attack on the Reich. By this criterion, the otherwise pathetically ineffective Zeppelin raids of WWI also had some success, since the British did develop extensive air defenses in Britain that did divert a lot of effort from the actual front. Hitler might rationally be going for a similar diversion now, and one can only hope the US military can fool the public on the cheap and not be too diverted. It does seem though that postwar, the diversion to defensive shelters and possibly ABMs will be major.