POD: The 1895 Venezuelan Crisis heats further, damaging relations between the US and Britain, due to handwavium. The actual invasion happens in 1904.
The key European alliances at the time are France-Russia and Germany-Austria/Hungary-Italy, and it's early enough in the German naval buildup for the Kaiser to dial back and mend fences with the British. The most immediately relevant rivalries are Germany v. France and, ITTL, UK v. US.
Due to assorted minor effects that somebody else who knows the time period better can sort out, the British view shifts that the United States is no longer a friendly neutral, but rather more of a true neutral wanna-be-hegemon. In the interests of establishing dominance in the Pacific, the British form closer ties to Japan and work to solve tensions with the Germans, alarming the French; this causes the French to reach out to the Americans. When the Germans look to give the United States a bloody nose to break American power before it can become overwhelming, the British are talked into lending support - the might of the Royal Navy, the staging point in Canada, and perhaps minor land forces (though only to defend Canada). The Japanese are invited along, as well, with the US Pacific possessions offered as a prize in exchange for opening a second naval front.
Due to actual ongoing tensions with an European power, Teddy doesn't allow the US Army to remain quite so unprepared, even if the army is small by European standards. The invasion, centered on New England and the Great Lakes, bogs down rapidly; the front is wide, the (newly formed) National Guard units are highly motivated, and much of the terrain is either highly defensible or difficult to traverse. H.M.S. Dreadnought has yet to launch and the German navy is still fairly weak, so the US Navy is able to function well in a purely defensive role, denying the British complete local dominance.
This has incredible potential to gradually devolve into a general Great War. The Russo-Japanese War may or may not be butterflied away, but the underlying tensions remain, and France is still quite leery of Germany; the French may decide that, with the Germans so committed overseas against their buddies and the British Navy tied up, this is their chance to strike. Multiple separate conflicts emerge: United States vs. Britain + Germany + Japan, Germany + Austria-Hungary + Italy vs. France + Russia, Russia vs. Japan.