Really tough.
By 1870, the LAST thing Britain wanted was a continental rival in the Americas. The US had surpassed Britain economically by then, and by 1872 the UK was facing the reality of a United Germany in Europe.
1. The US and Britain enjoyed extensive, mutually profitable economic ties.
2. The US and Britain had commonality of language and - somewhat - of culture.
3. The last time they fought - 1812 - they had damaged each other badly for no appreciable gain. This would be worse in any new go-round.
4. They have no disputes from 1870-1914. The US is (mostly) isolationist. Britain is concerned with:
A. Maintaining the Concert of Europe
B. Forestalling the rise of a single dominant European power
C. Keeping the Russians from a warm-water port
D. Maintaining the British Empire
E. Defending the global trading network
Of those five points, the US has no real interest in points A - D (although it would be more inclined towards the British on all of them except D) and is in enthusiastic, active agreement with Britain on E.
People mention the Venezuela/Guyana thing in 1896. This was barely even a footnote at the time; you need serious screwage to make a war of it.
The real goal of this thread is to fight a war between Britain and America and have a fun discussion of the relative merits of both sides' predreadnaughts. While I don't dispute the fun of that, it is implausible. Neither Britain nor America planned to fight a war with the other, and the force structures and deployments of both powers reflect that. If the United States had seriously believed a war with the UK was possible, it would have a much larger army in 1876, and it would have been deployed far differently, for example. The British would have had a much bigger naval base at Halifax, etc. Plus, of course, this doesn't happen in a vacuum; how do other major powers react, are there two sides, etc.
And of course there is the inevitable nationalist chest-thumping that this discussion would generate on this board.
Mike Turcotte