Any grand late 19th century war in Latin America (and the Americas as a whole) will be a US-UK proxy war, and one which the pro-UK side gets weaker and weaker as the years pass. It would essentially be a struggle between the existing British domination of Latin America and the rising American domination of Latin America.
But if we need this to happen, then we have:
Entente:
British Empire - Rulers of the Caribbean, global reach of the Royal Navy
France - Caribbean colonies, French Guiana
Argentina - Economically linked to Britain, anti-Brazil
Canada - British dominion, enough said
Newfoundland - See above
Colombia - Wants Panama back, stands to gain a lot from Entente victory
Central Powers:
United States - Economically the strongest world power, strong navy, TTL decent sized army (due to threat from the British)
Brazil - US economic interests, anti-Argentina
Central American states - US puppets essentially thanks to United Fruit, the Canal Zone, etc.
Mexico - Central government backed by the United States, opposed by many rebel factions, a complete mess really
Cuba - US puppet
Dominican Republic - US puppet
Haiti - US puppet
Neutrals:
Chile - Could be persuaded either way thanks to links to both Britain and the UK
Peru - Rival of Chile and Ecuador, but once again could be persuaded either way
Bolivia - Rival of Chile and Paraguay, could be persuaded either way
Uruguay - Influenced by both the UK and the US, no real way to profit from the war
Paraguay - Rival of Bolivia, but no real way to profit from the war
Venezuela - Claims on British colonies, but many economic links--Venezuelan oil is an important commodity--could go either way
Ecuador - Rival of Peru, could go either way
If I wanted a hellish war in Latin America, I'd assign Chile, Ecuador, and Paraguay to the US side and add Venezuela, Uruguay, Peru, and Bolivia to the British side. This makes it somewhat evenly matched but sooner or later the United States will end up dominating and its supplies and finance to its allies will help grind down the opposition.
Even if we throw in a Japan firmly dedicated to the Entente instead of just opportunistically seizing colonies, things would pretty much be the United States gets a free hand in reshaping Latin America in the postwar. Also, Canada gets smashed but that's just your typical 20th century Anglo-American War.
End result is likely Paraguay, Chile, and Ecuador getting maximum claims on their rivals, friendly rulers being installed in pro-British countries (likely long before the war ends, thanks to opportunistic caudillos), and Canada losing at least British Columbia, if not everything west of Ontario. The British and French Caribbean gets annexed to the US, except for bits of Guyana (partitioned between the US and the pro-US caudillo in Venezuela) and French Guiana (which goes to Brazil). The naval war will be key, since that's how supplies will pass to Latin America and Canada, but there won't be a lot of British or American forces deployed there (some Americans may fight some Mexican rebels, while the French and British will defend their colonies). American participation in Europe will be strictly sending supplies to Germany, likely escorted, or helping blockade Britain. The main fight of this Latin American war will be in the Rio de la Plata basin, but will likely in Central Powers (Brazilian) victory.
Thank you very much. But do you think it can be done with a POD between 1770 and 1830?
The easiest way for this to happen is American-British rivalry, or at the very least one which exists after the American Civil War due to British support for the CSA (successful or not). Otherwise the United States and United Kingdom are hard enough to draw apart to really have a massive proxy war in Latin America.