As we have frequently discussed here, although Britain remained neutral, there was a thriving business in supplying material for the rebel war effort. And many British subjects openly favored the rebel cause. In the early days of the Lincoln Administration, Secretary of State Seward suggested creating a crisis with Britain as a way to reunite the Union. Lincoln rejected the idea. In the Trent Affair, Britain prepared for war by sending 11,000 troops to bolster provincial militia. There was also serious, ongoing concern over cross-border raids into the US by rebels based in Toronto and Montreal. There were calls for Lincoln to send troops across the border to stop these raids.
So there was at least some reason for Britain to be concerned at both the outset and during the rebellion. As a result, Britain enhanced border security, stepped up military training and deployed some naval forces.
As to the post-war situation, there was friction between the nations, partly because of the Fenian Brotherhood- Irish Nationalists who conducted raids against British installations. Many Fenians were veterans of the recent rebellion.
As I recall, the British defense was largely local militia and regulars already stationed in what became Canada. The US government took steps to prevent Fenian raids. There were, no doubt, remonstrances in Parliament and threats of retaliation, but I don't recall any reinforcements sent from elsewhere in the Empire.
As president, Grant made what Britain considered inordinate demands as reparation for what unofficial support British subjects had given the rebels. There were discussions between the governments of settling these demands by ceding the remaining British North American provinces- modern-day Canada- to the US. Eventually, negotiations led by John A. Macdonald put an end to these demands and set the stage for eventual Canadian independence.