What if the Treaty of 1818 resulted in a Bir Tawil Situation?

What if the Red River region ceded to the United States as a result of the treaty of 1818 was not recognized by the British. Would the two nations go to war over the stretch of land? If they didn't, would the US cession to the UK become a Bir Tawil situation?
 
Nobody wanted war in 1818. If Britain didn't go to war when Jackson hanged two British guys in Florida without trial (one of whom was probably innocent) they sure as hell won't start a war over the Red River.

Either Britain cedes it or it goes joint ownership like Oregon.
 
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