However much it offends the American ego, the USA was pretty much irrelevant to the European wars of the 1800s.
IMHO the British would treat a DoW from USA vs Bonaparte much as they did the "Quasi War" of the USA vs Republican France... nice to have but no practical effect in their on going struggle with France (however it is governed)
This irrelevance is best demonstrated by observing that the direct opposite. i.e. the OTL war of of 1812, was a very minor sideshow for the British.
IMHO US support against Napoleon changes almost nothing up until 1815
... and even in that year would have had no significant effects to the end results.
Despite its early successes against RN vessels, by 1814 half the ocean going USN had been destroyed by a small percentage of the RN. Inversely adding the USN to the RN in 1812, would do very little as there would be very few targets for the US raider/privateer tactics.
Similarly while some British flagged merchantmen had been captured in OTL (though many were retaken)
US flagged merchantmen were decimated to the point where US investors refused to finance further voyages.
However due to the vast difference in the sizes of their merchant fleet , while this exchange damaged the US for the UK it was only around break even.
Inversely, it is true that up to 1812 US raw materials and US merchant vessels had been useful supplying British bases and armies meaning that would have continued to be advantageous.
However when these became unavailable the British found adequate alternative sources e.g. Egypt.
A US alliance with Britain would at least have stopped some of the the US traffic to Bonaparte
though IMHO some Yankee traders would have continued to smuggle extensively (money talks).
On land, Initial US aggression against Canada had been repeatedly repulsed from purely local resources
so there was little impact on the Peninsular War. Reinforcements from Europe were only available after Bonaparte's first abdication, meaning the British could only take the offensive after mid 1814.
And that diversion, supposedly in time of Peace in Europe, is where this POD might have some impact.
Without the extra troops in North America, Britain would have been able to respond better to the 100 days.
Some of the battalions sent west in OTL would have been demobilised iTTL but even these could certainly have been remustered more quickly than shipped home.
Not that the result would have changed of course, but perhaps the mechanism.
Given some extra veteran troops and better subordinates, perhaps Wellington could have taken the offensive in the afternoon of Quatre Bras and moved against Bonaparte's his left flank at Ligny
(Effectively swapping roles with Blucher compared to two days later at Mont St Jean)
Given this degree of insignificance, I do not foresee any mid term consequences
and long term America has to become strongly anti British to become the dominant world power.