Considering these are major languages of Spain and have at times enjoyed some official recognition and I presume people moved from these areas to the Americas, why have they failed to take root in the Americas?
The Spanish missions to the New World were almost wholly Castilian affairs. I doubt that one in a hundred of the sailors on those voyages came from Aragon or Andorra.
But the sailors on “those voyages” (the early voyages of exploration?) are not the main source of Hispanic language, culture and identity in South and Central America. They were greatly outnumbered over the next centuries by priests, soldiers, colonial officials, and civilian colonists from all over Spain and Portugal and beyond. Very many of whom were Basque, for example.
I get the impression that the colonial officials were mostly appointed by the conquistadors themselves, at least at first. Were there a lot of Basque missionaries and settlers?
Yes, and officials. They were well-represented in the ruling classes in some countries. A very significant percentage of people in Chile, Cuba, Colombia are of Basque descent. At least one viceroy of New Spain was Basque. Lots of Spanish Jesuits were Basque.
There may have been a leveling effect and/or influence from those in power. Think about it this way - how many Lowland Scots, Irish Gaelic, Scots Gaelic, Welsh, and Cornish speaking areas are there in North America compared to the number of people who are descendants of speakers of those languages?
Imagine this. You’re a Catalan speaker from Barcelona. You settle in San Francisco next door to a Galician speaker and across the street from a Basque person. The language of the Government is Castillian. Which language are you likely to try to concentrate on when going about in public? Which language will you encourage your children to use?
Until quite recently, there were more native speakers of Welsh in the US than in Wales.
Indeed, the King’s service did include a large Basque contingent, and there were a goodly number of gallego traders and merchants, and there was clergy from both (however, most Catholic clergy don’t leave as many descendants per capita as the laity 
). But as mentioned, the official standard language of conducting business was Castillian (or Latin, within the Church). Regardless of what fueros may have applied in Galicia, Catalunya or Navarra for their populations, the Empire was run from Castilla and Andalucía.
There is a large gallego community in the Americas today, but most of them are from families that emigrated after 1800.
Well, a lot of those Basques were Castillian. Not Castillian as in “from the Spanish regions called Castille”, but as in “from what’s nowadays called Euskadi, which belonged to the Crown of Castille since the middle-Middle Ages”.
Thing is, in order to get people speaking a language continuously for generations, you need a large-enough group. As robert_columbia has explained, people were much more likely to end up traveling and living with people from ten different regions, including some from Italy or Portugal, than in a group in which most were from the same area. Add that Hispanics* consider it very impolite to speak in from of someone in a language that someone doesn’t speak (unless that someone has specifically asked you to do it so they’ll learn the language) and you have those who were bilingual automatically switching to Spanish.
- current Catalan politically-motivated behavior is an aberrant. Also note that, while many Catalans will switch to Spanish only if money is involved, they’ll happily switch to any other language if they happen to speak it.
But were the any pockets of Basuqe, Catalans or Galicians in the Americas and did any survive to this day or say till the 20th century. I mean Argentina still has Welsh speakers, so please gimme a couple of Catalans?
That’s interesting. How recently?
Well, not that recently. It was prior to the Welsh Language Act of 1967, certainly- in the thirties, perhaps. I’ll have to see if I can find the book I read it in.
Also, keep in mind that most governments, until quite recently, had no problem with forcing people to speak a particular language. And with the very large number of Mestizos in Latin America, Castilian would be the prestige language. Promoted above all the native languages and other Spanish dialects.
Also, note that something quite close to Gallician is spoken in one SA country. I’ll leave the answer to that as an exercise for the reader.
I had to look that up. I had no idea that Portuguese and Galician are considered by some to be varieties of the same language!
Well, what do you know.
Monolingual Galician speakers living in Latin America would be encouraged to learn Castillian to survive. Bilingual Galician/Castillian speakers would be encouraged to use Castillian in day to day life over Galician, and perhaps they would bring their children up as monolingual Castillian speakers, because Galician did not have sufficient utility over the cost of teaching the kids to speak it.
Also, the Spanish language itself is sometimes called (in Spanish) castellano instead of espanol, acknowledging that there are other lenguas espanolas (Spanish languages). I see this as similar to not calling the English language “British”, because there are other British languages, such as Welsh, Cornish, Scots Gaelic, and Lowland Scots.
I’m no expert but I find this assertion a bit suspect. First off, there are only about 2500 Welsh speakers in Canada currently versus about 600,000 in Wales. However, in the early 20th century the number of Welsh speakers in Wales was around one million. Wales also never had the level of migration to the US that Ireland or Scotland had.
Excuse my ignorance, but are Catalan and Galacian separate and distinct languages, or are they dialects of Spanish, comparable to British English vs. American English or Australian English?
Both Catalan and Galician are separate and distinct languages from Castilian Spanish.
It depends on whom you ask. The decision is as much political as it is linguistic. Galician is probably more difficult for a native Castilian speaker to understand than Catalan is. But I would defer to our native Castilian speakers on that one…
Remember, Catalan is very similar to Occitan, and Occitan is not (for political reasons) recognized as a distinct language in France.
JKellyMap: Look at a map and it’s not so surprising! 
They’re distinct dialects with their own formal constructions and spellings, but Catalan is recognizably a close relative of Castilian, as are Galician and Portuguese.