The first thing to note about this scenario is that it needs a formidable 'deus ex machina' that the OP has not worked out or at least not mentioned to make it work. Consider for a moment: what possible threat or inducement could the USA apply to Britain to make it give up a valuable colony even if it was not part of the Canadian Confederation? There is nothing the USA can do to make the British give up BC, thus the only way this can work is if the British, the Canadians, the Hudson Bay Company and the colonists in BC itself want to get rid of it and frankly why they would want to do this is not obvious.
So accepting for the sake of the argument the flawed premise above. The first thing we can say is that in this scenario the British, knowing American aspirations in 1866 are highly unlikely to amalgamate the Colonies of Vancouver's Island and British Columbia. Thus we end up with a new American territory (British Columbia) that has a British Colony with a large naval base and by far the most powerful war fleet in the Pacific just a few miles off shore. On the basis of this geography American BC loses all of our BC's fishing grounds on top of this the gold is played out, it is cold and a long sea journey from the East Coast. It is thus likely to remain just as isolated and underdevoped territory as Alaska was in OTL.
Next there is no treaty of Washington and no great impetus for it as there was in OTL due to the Alabama claims. This means several things first, the principles of international law regarding neutrality and commerce raiding which the Alabama Commission gave rise to will not come into being. Second, the Halifax Fisheries commission will not resolve the joint exploitation of the Grand Banks and a festering sore in Anglo-American relations will continue. As a result of this the numerous OTL trade barriers between between the USA and Canada will continue (unlike OTL), there will probably be even more. It is not unlikely that. American ships will be denied access to the Atlantic via the St Lawrence by the Canadians (rather than the British). This will greatly retard the development of the USA regions south of the lakes. It might (or might not) enhance the development of Kingston as an international transshipment port (i.e. goods removed from American ships, import duties paid and loaded on to Canadian ships to load ocean going ships at Montreal). On top of all this without the treaty of Washington the dual occupation of San Juan island as a result of the Pig War.
In OTL the Russian Empire offered Alaska to Britain Before they offered it to the USA. In this scenario there is a very good chance that the British or at least the Canadians will want to buy it as it gives Canada access to the sea and Vancouver's Island.
The borders of BC to the east were not defined to my knowledge and would have to be agreed before handover as the British are no more fools than the Americans. It is thus likely that this BC could be a lot smaller than our BC. There is also an interesting discussion to be had as to the northern border.
So in conclusion, the POD is unbelievable but if the USA did get BC then Canada gets Alaska. BC remains underdeveloped. The USA's lakes region is less developed than OTL. Anglo-American and American-Canadian relations are poorer than in OTL. The channel between BC and VI becomes a potential naval flash point