Revenge? Look at Alsace Lorraine, or Danzig.
I suppose it's entirely
possible that that would take place, but equally one can look at - just as examples - eastern Poland, or most of old Austria Hungary, or Britain with America (no serious attempt to reclaim) or Britain with France after France helped hive off America (no serious attempt to punish) or Russia with France and Britain after the Crimean War, or Austrian attitudes to Prussia post-1864, or Danish attitudes to Prussia, or Dutch attitudes to Belgium, or plenty of other cases where there was no serious revanchism.
In fact, as a rule, successful wars of independence then don't seem to provoke subsequent reannexation wars, and successful interventions don't tend to provoke subsequent punishment wars - unless, of course, there's other prevailing geopolitical drivers towards it.
And in the case of a Union which has lost the Confederacy, it will absolutely
not want to fight Britain and the Confederacy at the same time again. That war would be the platonic ideal of a multifront war - attacks being possible on the South, the North, the East
and the West.
Instead, it would most likely want British help against the Confederacy, building on a common heritage, while also not wanting to reannex the South as that would result in some rather nasty guerillas, and focus on getting rich.