The Antarctic would be a bit warmer without that circumpolar current--Chile would merge seamlessly into the Antarctic peninsula, perhaps by way of a long arc of land through the Falklands. The Isthmus which would be warmed on its east side by the Brazil Current. Conditions along the Antarctic coast might resemble Norway as far as parts of OTL Antarctica--though the projection of the Land Bridge around the Scotia Arc would direct some warmth away from the continent itself. The west coast will be much colder, though. The disruption of the Antarctic current might also warm up patches of the coast closer to Australia, too. Kerguelen might be a bit nicer.
The natives of South America continue along the isthmus, disrupting ecology for a long distance--penguins and seals will suffer a pretty bad population hit. Maybe something analogous to Inuit culture will emerge there, with people riding or kayaking across the edges of the continent to new fishing grounds.
The first notice of this new land will be Spanish or Portuguese navigators who find the Straits of Magellan, which make a semi-convenient crossing to the Pacific, and then notice these new lands. The continent will get a few Spanish and Portuguese missions and outposts. The Straits of Magellan will be fortified heavily by Spain--they are now the only way to get to the Pacific from the Atlantic. Indeed, I imagine that wars will be fought over them, as France and England attempt to take the straits from Spain by force. The Spanish might try to take over South Africa at some point to gain a total monopoly on maritime travel between Europe and Asia. Even if they don't, the Straits will be a lucrative source of income for centuries.
The Pacific becomes the next big question--Spain and Portugal are going to be the only serious powers there until the 18th century, when France and Britain and the Netherlands show up. With Spain's dominance over the Antarctic Isthmus, the other European powers will take an earlier interest in South Africa and Australia as supports for their east Asian interests. Polynesia will fall into some irrelevance, as it's "on the way to Asia" only if you start from Peru--the only people who will care about it either way will be Spanish missionaries.
North America will get even more investment from England and France and the Netherlands in their search for a Northwest Passage. By 1800, it should be clear that there is no such thing. Panama, Nicaragua, and Mexico are already Spanish, though Britain might try to take Panama and Nicaragua and set up a railway later in the century.
Will they succeed?
That depends on the state of the Spanish Empire. I imagine that Napoleon is butterflied away, so there might not be a Peninsular War, and Spain manages to keep a tight hold on her colonies. If no similar disaster happens, the Spanish Empire gradually turns into a Spanish Federated Kingdom, ruled by the King of Castille, Aragon, Mexico, Argentina, the Antarctic, and Peru--Spain is first-among-equals, but all are equal provinces.
In total, the big winner compared to our timeline is Spain. Whoever colonizes Australia first (Portugal, maybe?) is the runner-up.