Sorry for the delay in responding to this - it's been a busy few weeks.

This is in no way a comprehensive answer, but I did want to share a brief anecdote that shines some light on how at least some British citizens saw the War of 1812. A little more than a month after the victory at Waterloo, Bath's Sydney Gardens (an outdoor pleasure garden akin to the more famous Vauxhall in London, threw a celebratory "Grand Gala" to benefit the Waterloo Fund. It featured many of the elements that would become standards of Waterloo commemorations, including themed (and patriotic) musical performances and fireworks. The fireworks featured Napoleon confined on board HMS Bellerophon and, crucially for this question, the national emblems "of France and America overpowered by the Royal Standard of Great Britain.” (Bath Chronicle and Weekly Gazette, August 17, 1815, p. 3) Now, at least part of this may have been an attempt to wash the bitterness of New Orleans from the mouths of Bath's citizenry with the sweetness of Waterloo, but it is telling that America was included. To me, this indicates that, at least for some, by throwing their lot in with Britain's old enemy, the USA had become linked inexorably to France, to the point where a victory over one was a victory over both.

That said, I don't believe this was a particularly common association, or one that lasted, as this is the only celebration I've ever come across that included mention of the Americans (and I've read quite a few accounts of Waterloo celebrations). As for British leadership, the traditional reading is that they sought to curtail US territorial and naval expansion, rather than recover territory lost in 1783. (see, for example, Tory Bickham, The Weight of Vengeance: The United States, the British Empire, and the War of 1812 (Oxford: Oxford University Press, 2012); Jon Latimer, 1812: War With America (Cambridge: Harvard University Press, 2007); for an overview, see Duncan Andrew Campbell, "The Bicentennial of the War of 1812: Reconsidering the 'Forgotten Conflict'" American Nineteenth Century History 16, no. 1 (2015): 1-10.)