Oh, I was so hoping you or e of pi would answer this question!
Well, it's a bit of a long and complicated story, as most things are. I know that even in the 19th century Canaveral was identified as being pretty good for this, given its famous use by Verne, but on the other hand that was partially because there was an organization similar to the one in that novel operating in the area at the time (probably because it was in the middle of nowhere).
Middle of nowhere, one key criterion.
... you can see that the Cape sticks way out from the coast. Not a big deal with things like the Space Shuttle or Saturn V, or even the Atlas, but early on, when they were just launching relatively small rockets, that gave them a pretty decent-size clear area for launches that had no chance of becoming unsuitable. The next farthest-south area with anything close to the same launch azimuths available looks to me like West Palm Beach, which was, like Miami, already a fairly important city by the time the Cape was created.
It's an open question whether it would be ITTL, but going farther south than Canaveral in Florida means that now the Bahamas are due east; it is a good idea to avoid the launch tracks going directly over them--near them is OK, great in fact for downrange tracking and basing boats and planes to scramble out to investigate wreckage from an aborted launch. But not so good to go right over them; Canaveral is about 2 degrees or 120 nautical miles north. The launch track of a "due east" launch would start to veer south of course so I'm not so sure it avoids going over the Bahamas OTL, but anyway the more northerly site of Canaveral reduces the problem.
Additionally,
according to the 45th Space Wing's official history (they're the people who run the Air Force operations at the Cape), it was logistically convenient....
A good trick when one has tried to meet the "middle of nowhere" criterion.
...ITTL, I suspect a site in the Bahamas would be considered but rejected due to the (admittedly small) logistical burden created by having to ship in every test vehicle, instead of being able to transport some supplies by road or rail.
I've been fooling around with Google Maps and I identify two possible candidate islands:
Great Abaco Island--specifically the middle part of it, centered on a large bay on the east which oddly enough Google Maps does not name, between Lynyard Cay and Winding Bay--the capes defining it seem about the right size for launch complexes. To match the Kennedy/Cape Canaveral Air Force Station complex pretty much the whole central part of the island would have to be annexed to the station. Then of course OTL I'm sure quite a lot of the support that these complexes rely on comes from corporate facilities off the site but in the neighborhood, so the whole island and parts of neighboring ones might wind up being essentially part of the launch complex.
Mayaguana Island is the easternmost island of the chain/legal entity of Bahamas; it is near Grand Turks and Caicos but they are well to the south. I'd think this would give the clearest range of launch azimuths out of any US possession ITTL. However the island is small, less than half the area of OTL Kennedy Center itself (without adding in CCAFS) and quite isolated. The OTL population is under 300 so getting eminent domain would probably not be a big problem, but it is 450 miles east of Palm Beach. Definitely the middle of nowhere, but the logistics would suck, and the limited space would impose some limits on what could be launched there.
One does not yet know what sort of space program TTL USA would go in for though. OTL, in the 1960s designers were looking ahead, in the 70s and beyond, to making launchers capable of putting a
million pounds of payload into orbit!
It sounds a bit less Cecil De Mille when you convert to metric and use tons; that's "only" 450 metric tons, "only" 4 and a half times the capability of the Saturn V.
Is that too much to ask?
Picture a rocket with 22 of the Saturn V's 5 F-1 engines--or a mere 20 or perhaps 18 of the upgraded F-1A. Or something like six space shuttles launching simultaneously....
I don't know if Kennedy Space Center was meant to be able to handle those monsters (which were also supposed to be reusable by the way) or whether it was assumed some new complex would be set up somewhere.
So, if ITTL the American space program is more modest, they might still plan for a margin of optimism that would demand rather more elbow room than they need immediately. Being based on an itty bitty island puts some definite limits on those ambitions.
One interesting alternative possibility I just thought of might be Wallops in Virginia. That was created by NACA in 1945 for high-altitude sounding rocket and balloon launches, and has since been adapted into a space launch facility (albeit a relatively minor one). Virginia of course is quite convenient to NACA's pre-NASA center in Virginia (Langley), so it's plausible that if some similar organization is created here that it would also have a test center in that location. If it were tapped to lead a civilian space program, like NASA, or more pertinently the Naval Research Laboratory (which ran Project Vanguard), then it might choose to use the existing facilities at Wallops rather than develop new ones at the Cape (assuming that a test ground wasn't established at the Cape, or that there was some reason like the eternal Air Force unwillingness to work closely with NASA for using those). Wallops has a latitude similar to many other launch centers, such as Baikonur, Jiuquan, or Tanegashima, so it's not impractically high up, either, if not as optimally located at the Cape or Kourou.
Again it depends on ambitions. A high latitude site does not impede missile development in the least; no one cares how high the inclination of a missile's orbit is; it just has to complete a suborbital path and reach its target and you can launch from anywhere to achieve that. So the "elbow room" criterion applies most strongly, not only for public safety but also for military security and secrecy.
I can go you one better, off the top of my head--the Jersey Pine Barrens. That's where Fort Dix and Lakehurst Naval Air Station are OTL, on the northern fringes of it anyway. For whatever reason there's this desolate stretch of coastline way up between Philadelphia and New York City, quite accessible to some of the biggest industrial concentrations in the nation yet with big stretches of land easily cleared of all innocent bystanders.
OTOH if reaching useful orbits is the criterion, I'm a fanatic for low latitudes. Mayaguana seems the limit there; it's six degrees south of Canaveral.
But Canaveral of course has proven it is good enough. As to be sure so have the higher latitude sites you've mentioned.
I wanted there to be a NACA equivalent ITTL--in fact, what I wanted was a NACA on steroids, at least partially under Peace Department supervision. This would give the PD something real to do. My notion was that a political deadlock between people who wanted to build up the US military along OTL lines (for reasons both of imperial ambition and corporate pork) and the stronger pacifists of the timeline--make developing new airplane models adequate to defend the USA in case of need the responsibility of an agency under the Peace department, so there are checks and balances between prudent defense preparedness and "merchant of death" boosterism, and also charge it with assisting American aeronautics firms competitive through aggressive development of advanced technology; this in lieu of these firms expecting a chance to bid on juicy Army and Navy orders--the services get a very thin gruel of token numbers of fairly modern planes, and funds to keep a flying circus of obsolete models going, so their aviators have some flight experience and have to share brief tours with the latest stuff--in case of serious war threats the services would of course be expanded with orders of the latest stuff. This way the US has some preparedness for serious war should matters come to that, but avoids the high costs of maintaining a strong force and the implicit threat that conveys.
Such was my suggestion, but it didn't spark any interest. The Department of Peace, we are told, remained a moribund and irrelevant political sinecure after Jane Addams left it; it did not acquire an air R&D arm, whereas I can't imagine the US Army Air Corps could have ever amounted to much, while even the Navy is probably consistently smaller than it was in any given year of OTL (before the WWII buildup I mean--after 1937 or so the USN of TTL would be increasingly dwarfed by the one of OTL).
All of this means--very few tax dollars supporting the various aeronautical firms; they all have to make it or break on private, commercial business (or war materiel sales to overseas customers--but the Great Powers will have their own and their client's markets sewn up, so that leaves slim pickings).
Therefore the infrastructure that is close enough to the needs of a rocketry program would be mighty scarce in the USA.
This is my hidden agenda for the Peace Dept super-NACA; I wanted a more robust, cutting-edge US aero industry without the OTL war-mongering, and I wanted a government agency besides the Navy capable of taking up the mission of space exploration.
I suppose there is probably a NACA of some kind ITTL--maybe. OTL it was a Wilson Administration thing, tied to the scare and eventual involvement in the Great War. God knows it operated on a shoestring budget OTL, so it won't take much tax dollars to have something like it by the way here, but again the government's involvement and concerns with aeronautics are less here; if the alt-NACA is defunded in proportion to the military, it will probably drop dead of anemia. With the OTL one a war-boom sort of thing I'd expect its ATL cousin to have been something cooked up by the Lodge administration--and axed when he finally left office by the anti-war coalition that took over. (This was the juncture where I hoped the Rube Goldberg Peace Department NACA would come into being). So there might well be nothing of the sort.
If there is, I guess it would have an HQ somewhere in range of Washington, but not necessarily in DC. I've personally resided at Langley AFB and I know that it is a good half-day's drive from DC, not exactly next door--but it is possible to get from one to the other within the same business day. (In the early 20th century I don't know that one could shuttle back to one's starting point again also in that same day, and still have time to get any business done).
So I don't see the hand of inexorable fate at work in locating the main offices of NACA in southern Virginia rather than say in northern Virginia or in Maryland or even in say the Pine Barrens!
