Clearly the only solution to this hypothetical war is to war game it.
Does the setup for said intellectual exercise include the historical reality that the men in the US government and in the US armed forces fought and won a complete victory (unconditional surrender) in a total war stemming from an existential political crisis, on a continental scale, with mass mobilization and related actions in almost every ocean, and against a Western peer competitor, all in a period of 48 months ... While the men in the rebel government and forces lost said conflict?
And that those in the posited British government and forces, on their own, never faced a peer enemy in the entire period?
The obvious issue is that the evidence is in hand for the performance of the US and rebel nations, as well as - for that matter - the other major actors in North America at this point, the Mexican, French Imperial, Russian Imperial, Spanish Imperial, Comanche, Dakota, Dominican rebels, etc. as well as minor actors like the assorted provincial and colonial governments of British North America, the Fenian irregulars, etc.
What is not available, however, is any real world evidence of the performance of the British Imperial elements against Western peer enemies in the 1860s, other than second-hand at Ridgway and Fort Erie (where the Canadian provincial forces, under British strategic direction, failed miserably at the operational and tactical level against irregular non-state actors, despite four years of supposed mobilization against the very threat they faced with the Fenians and as many years of training under the aegis of the "professional" regular British Army) or against non-western enemies like the Chinese in 1859 (where the Royal Navy failed miserably, losing multiple warships in action for the first time since Lake Champlain in 1814) or in 1860, where they required an 18,000 man expeditionary force, including 7,000 French troops, and French naval support to achieve anything against the Chinese.
The 1854-56 conflict (24 months, inconclusive result) of the Allies against the Russians doesn't apply, of course, given that without the French (who outnumbered the British on the ground in the only significant campaign by 100,000 to 28,000, according to the British official history) the Turks (45,000, also a much larger commitment) and even the Sardinians, who managed to deploy an expeditionary force of 15,000, more than 50 percent that of the British Army), the British alone would have been laughed at by the Russians.
Then there's the minor historical reality that ironclad navies were not war winners in this era in conflicts between Western enemies (if ever) as witness the courses of the Austro-Prussian and Franco-Prussian wars, much less the Franco-Mexican and Spanish-Dominican conflicts. All four were lost by the power with maritime superiority or even supremacy; the Austrians won the only ironclad squadron action worth the name fought in European waters, conclusively, and still lost their war against the Prussians and Italians. The French navy was useless in any real terms in 1870-71, and the Mexicans and Dominicans had no maritime resources whatsoever compared to the "European professionals" of France and Spain, and yet both European imperial powers lost those wars in the 1860s as well.
The historical evidence is such that:
A) Ironclad navies had little to no influence on conflicts between Western powers in the 1850s and 1860s; and
B) conflicts between Western powers in the 1850s and 1860s came down to military power ashore, which in turn depended upon population, economic resources, and mass mobilization in the theaters where the conflicts were fought. The imaginary advantages of any "professional" army never made any difference on the battlefield.
The above is the historical record, repeated multiple times, in the very period under discussion, and in the real world of mud and rain and salt spray and the boredom and blood and terror of actual conflict, not in intellectual armchair advocacy.
So if a simulation came up with any sort of result that did not mimic those of the actual, historical, real-world conflicts that were fought and won or lost by actual flesh and blood in the same period, it obviously would be flawed.
Best,