OTL Argentina was discovered and claimed first by an English expedition that reached Cape Hoorn from the east in 1565. The area south of the equator stayed barely settled, though; most settlers went to northern Braseal to start sugar plantations instead.
During the chaos following the First English Civil War Argentine fell into quasi-anarchy. After Humphrey I had won the Civil War, he decided to take back his lands. In 1659, England (re-)started the colonization of Braseal and Argentine. 1662, the English (re)founded the city of Fort Humphrey at the site of OTL Buenos Aires. They started trading with the nearby Tawantinsuyu (the Inca empire).
Since the 1790s, many nobles and other opponents of the French republic fled from the latter sphere of influence, mostly to Britain or their colonies of Braseal and Argentine, where many get land from the British state, thus growing the population.
1832-36, the anti-British War of allied New Rome, Germany, German Atlantis and Canada took place. Unexpectedly, the Imperial Navy proved to be stronger than the Royal Navy, thanks to the numerous fights against Spanish Navy-in-exile. In the peace of Roma Nuova, Britain had to cede Haraldsland (OTL Newfoundland) to Canada and its lands in South Atlantis, which were divided along the Tropic of Capricorn: Braseal went to New Rome, the Germans received Argentine as a place for settlement (from now on, more often called Argentinien). Patagonia, being uninteresting for the victors, became independent Aymaria. In 1837, the first German settlers arrived in Argentinien. The British upper class was kicked out, while the ordinary Brits were assimilated with time. The capital Fort Humphrey was renamed Neu-Hamburg.