in 1812, British and American troops were shooting at each other. (About) hundred years later America was supplying England with weapons to fight the Germans.
When did they kiss and make up? Was it a gradual thaw or was there some particular instance where things were patched up?
It really took a long time and over the course of the 19th century there were several incidents that could potentially have escalated into Anglo-American wars. During the late 1830s there was a small border conflict in Northern Maine called the "Arstook War" which could potentially have escalated into a real war. The Webster-Ashburton Treaty was signed and the border was adjusted, so war was avoided.
Under the Treaty of 1818, the United States and the British Empire had agreed on joint occupation of Oregon territory (an area that included the modern US states of Oregon and Washington as well as what is now British Columbia). By the mid 1840s this had become a fairly pressing issue and James K Polk campaigned on a promise to aggressively pursue a territorially advantageous settlement in Oregon. However, Polk accepted a compromise and the Oregon boundary dispute was resolved.
During the same period, the British Empire opposed the annexation of Texas (among other reasons because many British officials saw an advantage in a counterweight to American power in North America and hoped that Texas would become a market for British goods that lacked the high tariffs of the contemporary United States) by the United States, which many Americans saw as antagonistic behavior. During the Civil War the British were officially neutral however they did allow the Confederacy to manufacture ships in England.
In 1895 the Cleveland administration involved itself in a territorial dispute between Britain and Venezuela, sparking an international incident that could potentially have lead to a war. However, by accepting American arbitration the British had recognized American hegemony in the Western Hemisphere. After the Venezuela Boundary Dispute Anglo-American relations continued to improve.
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It should also be noted that Wilson believed in being 100% neutral, meaning Germany could have bought arms and material from the US too but the german navy wasn't free to rome like the royal fleet could. So they couldn't protect the transport ships meaning they couldn't buy.