If things went to hell in the San Juan Islands you can bet that Palmerston, a very anti-US PM, is going to want war, though whether his cabinet would allow it is another question.
But in any case, I think the predictions of inevitable US victory are overstated. The US Army didn't exactly cover itself in glory during the OTL ACW, and the British had gained valuable experience in the Crimean War - especially in the field of storming fortresses from the sea (Bomarsund, Sveaborg, Azov Campaigns).
I think this is the most likely case:
1) War breaks out and Canada is vulnerable. UK speeds up military reforms and ships whatever can be spared for the 4,000-strong Canada Garrison, including Crimea veterans and troops fresh from fighting the Indian Mutiny. Naval reserves are properly activated and trained (following the Crimean experience). Army innovations such as short-service and localized recruiting will increase the manpower pool of the UK in the near-future, but these will take time to train. Admiralty quickly puts in large orders for gunboats and mortar vessels, supplementing their already-sizeable force.
2) Like the Fenian Raids OTL, the USA operates under the assumption that the Irish-Canadians will support them in an invasion. As such, they lead spirited but hasty attacks into Canada, and are easily handled by local forces. Worse still, Northerners make premature moves about making Canada a 'free state', which paralyzes Congress and critical time is lost.
3) Palmerston resolves to punish the US for daring to provoke the UK. A plan to annex Oregon country is formed, feelers extended to Mexico for the reconquest of the Southwest, and the UK contacts slave-states concerned about the effects of Canadian accession to their power.
4) Militarily, the British plan is similar to that practiced in Russia, and also to the one situation they have planned for in detail (which is capturing Cherbourg in France): rule the seas, capture strategic points, and bombard every US port into submission. This inflicts massive damage on the US without having to face their 'massive' armies (though the US wouldn't be able to field more than 50,000 for the critical first phase of the war, certainly not the hundreds thousands they did in the 'total war' of the ACW).
5) Winter 1859 passes and the US is ready to try again. The 1860 Election is imminent and the (newly-united) Democrats want victories to keep Lincoln out. So the US invades Canada again under Winfield Scott. Unfortunately for Winfield, he fights battles in the Napoleonic style - but the British have learnt from the Russians and are digging trenches everywhere to stop the American advance. US attacks stall and where they succeed, it's with massive casualties. Buchanan desperately needs somebody with experience of this new way of fighting, somebody who has seen the Crimean War firsthand. And this person is... George B McClellan.
6) On the seas, the US rightly sees the Russian strategy of keeping its ships in Sevastopol as disastrous, but they can't engage the British fleet one-on-one. The US Navy therefore disperses and disrupts British shipping: it is an effective tactic, but it means they are not around when the British Navy shows up. No matter - the US expects their Third System Forts to hold out, slow to recognize that the British have learnt much about attacking coastal forts during the Crimean War (since the contemporary focus was on Crimea, not the Baltic).
7) The British fleet arrives and promptly begins work. Not wanting a long war, the First Sea Lord (Lord Dundas, a veteran of fort-storming ops in China and in the Baltic) decides on knockout blows to the US' major cities. Boston becomes the first city to experience the wrath of the British fleet as a combination of gunboats, mortar boats, 10+ battleships and naval landings assault the unfinished Fort Warren and capture it within days, making even shorter work of the older Winthrop and Independence. With Boston now defenceless the British now bombard the city (like at Odessa during Crimean War) and kill large numbers of people, sparking furious yet impotent national outrage.
8) Newport, RI is next, the British indicating that New York will burn without peace. 18th Century Fort Adams stands no chance and another American city is blasted. Then Fort Hamilton in Brooklyn, Fort Richmond in Staten Island, and Fort Schuyler in the Bronx are levelled, the British causing as much damage as possible to the region. New York City itself is probably too geographically unfavorable for naval operations, so the British sink blockships in the East and Hudson Rivers. In an instant, trade flows to the US' commerce capital fall to a trickle. Blockships sunk at the mouth of the Delaware (Philadelphia) produce a similar effect.
9) Canada continues to hold out because McClellan is hardly going to be even more decisive than OTL. In any case, the British decide to repeat 1812, reducing Fort Monroe at the mouth of the Chesapeake and sailing upwards with an elite army of soldiers. If the US has any sense, it would sue for peace now. If it doesn't... Baltimore will burn, followed by Washington soon after.
10) The UK continues to sail up and down the coast, bombarding all major cities with the US almost powerless to do anything about it, save for the occasional steamship sortie which might achieve limited, but ultimately irrelevant success. Canada remains a quagmire for American armies, with each successful advance stalling before another line of trenches.
11) War exhaustion mounts, ruining Democratic chances for re-election; Lincoln is now President. Bad for the USA as this means the South will now secede and make their own peace with the UK, leaving the Union to face the British with the majority of its resources gone. The British dictate terms.
