First of all, let's talk about the place. I learned about the Welsh Settlements there while I was in Puerto Madryn, in Patagonia, Argentina. This is where they first landed, and indeed along the shorefront some of the ruins of their first dug-out homes still exist. It must have been tough.
Fortunately they expanded, and the town of Gaiman is the Welsh home in Argentina now. Indeed Diana, Princess of Wales visited this town and had tea there.
There is quite a lot on the internet about the Welsh Settlement of Patagonia, and it makes for fantastic historical reading.
Now as you can imagine, it's not a massively expanding language in the midst of Patagonia. However, the Welsh Language Project has been set up to promote the Welsh language and culture there.
As they have teachers themselves and are teaching Welsh there, if anyone is going to know how you might be able to learn Welsh as a foreigner in Patagonia, it's this group. I suggest contacting Catrin Williams - there's an interview with her on their site as well.