The Mississippi River was an international border at the signing of the ToP and the US had no control over the river south of the 31st parallel under that treaty as that portion of the river was entirely outside the borders of the US. After the Louisiana Purchase in 1803 and the Adams treaty in 1819 the river without question lay entirely within the borders of the US, which would weaken any claim of rights of passage elucidated in the ToP. In any case relations between the US and Great Britain were renegotiated in the Treaty of Ghent in 1814 and British rights to the Mississippi River were not included/renewed. This was not a mere oversight either; the British pushed to retain this right, but it was flatly rejected by the American side as a deal breaker. So with the various changes in the political boundaries surrounding the river coupled with the Treaty of Ghent, no, British subjects have no legal claim to sailing Mississippi without American permissions.