For this update, I'm going to repeat some of the feedback from above, because that feedback was detailed enough to constitute part of an update, and also because the update would be incomplete without also taking note of these points.
Thanks, GhostWriter [question was about size of my fleet]! Your flattery is humbling, but appreciated! I think I would only have had time to build one or two cruisers, and maybe some destroyers or subs (probably subs -- would have fit my thinking better). But since to that point I hadn't seen the British AI send out more than a few ships against me at a time (that initial set, which I captured, being the largest I'd seen), I figured a fleet of maybe 5 cruisers was all I needed to fend them off. I got the impression the UK wasn't willing to send a REAL fleet after me, nor either a REAL invasion fleet.
Thinking back, it seems to me that when I invaded South Africa itself, the war in Europe hadn't yet started, but it was very tense. However, probably by the time I finished conquering S. Africa (it did take quite a bit of effort, and several shuttles of troops overseas, since I was limited by transport capacity) Sept 1939 had come and the war was on. At that point, I believe I was able to join the Axis.
So once I had South Africa (I don't remember if I completely secured the whole thing -- seems like I might have had to continue holding off armies from the north, coming through Rhodesia), where from there?
My answer was to invade the French and British possessions of the Caribbean!
I spent some time securing these islands, as I recall, and it was during this period of time that World War II officially started, with the German invasion of Poland, which of course immediately launched conflict along the Maginot Line (which isn't historical, but it always happened in HOI 1, that I recall). This took some of the pressure off of Argentina, and I figured I was taking some of the pressure off the Germans, too!
All in all, I couldn't have controlled very much of Britain's IC -- enough to help me some, but not enough to greatly handicap her. But I do wonder (and I never "peeked" to see) how much I disrupted her commerce and/or naval operations.
Here's more feedback, to TheExecuter, about naval range, which he reminded me probably didn't exist. You're probably right that HOI 1 doesn't have naval range restrictions. But perhaps it was an "AI attention horizon range" that I was seeing in action. I found that once I took Ascension Island, that became the focus of British attention (and later, S. Africa) so that Argentina itself was secure.
So (feedback off), I think I kept the Royal Navy occupied in a very disruptive way, so that they could not focus on Argentina, and I would bet (just knowing how the AI assigns targets) they also couldn't focus as effectively on securing the Mediterranean and/or North Atlantic.
Taking over the Allied possessions of the Caribbean took a while, as I mentioned. I recall I did dive into Belize, too, at one point, securing that for myself. Then where?