Argentina, Chile and Uruguay have huge Anglo-Celtic populations, with Chile having 800,000 Anglo-Celts, Argentina having 800,000-1,300,000 Anglo-Celts and Uruguay having 120,000 Irish-Uruguayans (Unsure about the amount of British-Uruguayans).
To my understanding, they have influenced the region quite a but, introducing British sports, founding hospitals and schools, and running most of the railways in the area. Less influential than say the Spaniards or Italians, but still very influential.
I have heard lots of English-Argentine live in Buenos Aires, and also Welsh speaking settlements are in Chubut, with lots of other Anglo-Celtic communities (Including Irish, Scottish and other Welsh and English in the rest of Argentina and neighbouring countries such as Chile and Uruguay).
What caused English, Irish, Scottish, and Welsh people to migrate to Argentina, Chile and Uruguay in the first place though?
This is a clarification question. What is the source for that number of anglo descendants in Chile? I’ve seen the number before but I have never believed it, the British and Irish communities nowadays seem fairly small, nowhere in the numbers or influence of Spanish, Italians or even Palestinians.
I will speak only for Argentina for it is the only case that you name that I have enough knowledge (I'm from Argentina)
In our Constitution we specify that we accept any person willing to live in our country and we will promote specifically Europeans settlers to come. Given our Latin roots, we were a natural destination for Spaniards and Italians, and today 90% of our population descends from those countries.
Now first I will talk about Welsh (and Scottish if I'm not mistaken) big community we have in our Patagonia. After our independence (and until the 1930's) , our main economic and political ally in the world was the UK and Durning the same time we were receiving their investment, we also acquired the southern territories (after Roca's desert campaign) Wich needed new population that could easily live there, so we needed Europeans from a similar terrain, and in the words of a descendant of Scots I knew in the South, the most similar place in the world to the Scottish Highlands, is the Patagonia. We made migration pacts with the UK to bring Welsh and Scots to the South and found new towns. Apart from the similar weather and unpopulated land, English speaking immigrants were reluctant to come to the big cities as they were highly dominated by Spanish (and a little bit of Italian too), the chance to live in a similar weather, speaking your own language and with big lands opportunities breed sheeps, was something very attractive to those settlers.
About the Germans we had something similar, we also promoted German immigration of course but the Germans not having settlements of their own (in colonies of their own) they had to settle in other language speaking new countries. They particularity choose the Sierras of Cordoba for the same reasons I explained before, unpopulated lands (note that Cordoba was not that unpopulated and because of that, the German community is not even as significant as the welsh-scottish community in the South) were they could found new towns to speak German on a daily basis and a similar weather (or that I heard about) to the Bavarian region. Something similar happened in the northern US if I'm not mistaken where there were towns founded by Germans for Germans and today an important part of the population is of German decent.
I hope that answers a part of your question.
A small follow up, but why does Brazil also present a large number of European migrants especially in the south, close to Argentina and Uruguay. Granted they’re not Anglo-Celtics but Germans, parts of the south of Brazil shares that common German and Dutch nationalism and Bavarian culture, look up “Gramado” city.
I'm no expert, but my understanding at least in Chile is that many Anglos came related to shipping and the merchant marine/navy. Remember that the UK had the biggest navy in the world. It makes sense that Brits would want to explore/do business in new ports/offer their services to burgeoning navies and merchant marines. Latin America was literally a gold mine for them.