Had the British intervened in favour of the Confederacy there was a very real and reasonable fear that that they'd lose Canada along with a very realistic concern that Russia might turn it into a global war because the US and Russia were on quite friendly terms back then thanks to the US support of Russia during the Crimean War and the Russian sale of Alaska to the US afterwards.
The British also would have shortages in the home isles worse than unrestricted submarine warefare ever did because, even with "those uppity cousins going at each other's throats" they were still exporting massive amounts of (mostly northern) foodstuffs to the British.
EDIT: I don't remember where I read that.
bintananth
Sorry I didn't reply further last night but in the midst of a multi-player gaming session.
For a hell of a lot of details may I suggest you see the start of
trent-war-possible-timeline - which is on the AH page of a USCW site. Not so much the TL itself which unfortunately seems to have stalled. However the author asked for and got a lot of discussion on the issue before the TL started which includes a hell of a lot of information. True the author is very confident about a British victory, see
if-they-will-not-meet-us-on-the-open-sea for what seems an earlier version of his TL, on the AH site. Love the impact on world history of the union asking for a Prussian representative at the peace talks.
To cut to the chase in 1861-62 the north, like the south, was still very much arming up for war. It was very reliant on imports from Europe and Canada in some cases, plus the fact that the south was in a similar but worst position. Later on it had its own developed gun industry but that's not the case here. As such a rejection of the British ultimatum, which Lincoln didn't realise would mean war, in which case.
a) The large number of rifles that the union had ordered from Europe wouldn't be arriving as their going to be subject to the blockade that Britain will be imposing. A lot of the Lee Enfield rifles might but they will be in the hands of Canadian militia. Note that also in this period a fair number of the US produced rifles were reliant on gun barrels imported from Britain.
b) Possibly even more critically to produce gunpowder you need among other items saltpetre and the current overwhelming source for this is India. It takes a long while to produce bed to locally supply this - 12-18 months IIRC and until that's done in large amounts the union will have a dire supply issue for powder. Some can be supplied from bat droppings in caves but again there are issues in setting up collection.
c) Money. Like any power the union depends on loans to pay for the war. Its going to face a serious problem as its now at war with the world's greatest lender. Furthermore while others are available, most noticeably France, their going to have a loaners market and also the union is a much worse risk so expect markedly higher interest rates. The government can try and get local loans, war bonds, but there will be limits to this. Matters are worsened as its primary source of income is taxes on imports and with a British blockade those are likely to drop considerably. Its also not going to be able to move Californian gold and Nevada silver very easily if at all as they came by sea - remembering this is before a trans-continental railway. Which apart from anything else means minting new coins is going to be difficult and in a war paper money tend to be unpopular. Even the chance of a war with Britain caused a run on the banks and an actual war is likely to make it worse.
Those are probably the most important factors but there are a number of others such as manpower - no further anti-slavery volunteers from Britain and Canada and migrants from Europe are likely to be significantly cut while the union now needs extra forces. [Yes you may see a lot of volunteers, especially with large areas of the economy in trouble but until they can be trained and equipped their an additional burden on the country.] Horses - a lot came from Canada and won't now. Even lead for bullets appears to be an issue.
In terms of the threat to Canada two points;
a) The war starts with winter as well which pretty much prevents any fighting in Canada until the thaw. Britain, in preparation for a potential war sent men and weapons to Halifax, many of whom continued overland to the Canadian provinces - now Quebec and Ontario to boost the defences both in terms of manpower and training/equipment for the local militias. If there's an actual war then this is likely to continue and any US attack in the spring is likely to run into serious opposition. Especially since their going to be facing British regulars, who have much better training than the union forces. Basically despite being increasingly equipped with rifles most of the forces on both sides were largely used as Napoleonic type infantry, assembled in line/bloc formation and firing at close range at other such formations. British forces were trained as skirmishers, able to hit individual targets at 400-500 years. As such anywhere there is sufficient range to use that ability union forces are going to suffer horrendous casualties. The smoke from gunfire is going to be a limiting factor as well but not before a lot of soldiers are killed. Especially if officers and NCOs are targeted there's a good chance that union forces are likely to frequently be crushed before they could close to firing range. There is a similar superiority in terms of artillery as the vast majority of union weapons are still smooth bore canon. Britain has rifled guns with much longer range and greater accuracy.
b) Also Britain can take the offensive on the eastern coast with attacks on union ports to tighten the blockade. The union has a number of fortifications but their overwhelmingly masonry forts, which are vulnerable to the 68lb canon that equip a lot of the RN battleships. Furthermore for budgetary reasons I suspect many of those forts have only a fraction of the guns and gunners that they were designed for. Which is before I believe a number of the heavier guns were transferred out to boost the defence of Washington.
Note that such attacks will not just cause material damage. Its going to prompt a lot of local governors and mayors, as well as merchants and ordinary people to want additional protection. Which will tend to draw men, artillery and powder especially from other fronts.
As for food supplies to Europe they will be disrupted because no US blockade runners will want to carry a heavy but cheap cargo like grain while British flagged ships won't be coming of course. As such it will have to depend on 3rd party shippers. I very much doubt that Lincoln will risk forbidding exports of grain so it will still occur. Note also that once that grain leaves a US port, and especially once its passed to a merchant in Europe the US has no say on where it ends up. As such British merchants can easily buy such grain or seek from other sources.
In terms of Russia don't rely on it. The country was shown how backward it was during the Crimean War and the Czar knows it needs a long time to modernism so it has zero interest in another costly war in Europe it can ill afford. That was the reason why it sold Alaska after all, because it wanted the cash for internal development. Also like other powers it was upset by the union actions in the Trent seizure. The Russians did send ships to the US during the civil war OTL but that was because during the crisis over the Polish uprising there was fear of war with Britain and France so they wanted their fleets in a safe neutral port where they couldn't be attacked.
In terms of raiders the US can try but:
a) privateering is now illegal by international law so any such ships will have to be formally USN ships.
b) Especially if much of the current USN, on blockade duty is seized/sunk by the RN when war starts the US will be short of men, gun and possibly trained sailors for such ships.
c) Unlike the US Britain has a world-wide collection of bases along with representatives in many big neutral ports. Coupled with numerous telegram lines that means that a raider that enters a neutral port, either to hand over a prize or gain supplies could well find RN ships waiting for it when it sails again. As well of course as a very large navy including many smaller ships that can hunt down a merchant ship converted to a raider.
d) If necessarily, as its less than 50 years since the Napoleonic wars have ended, Britain can restore a convoy system, at least in threatened areas.
Anyway sorry this is a long post but a lot to cover and more details in the 1st if those links.
Steve