Regardless of the gold rushes, it would be a naval affair, with seizure of the territory by marines occupying Sitka/Novo-Arkhangelsk. The 'border' between British North America (no 'Canada' as yet) was only sketchily defined, and barely explored much less a place for armies to tromp through. It wasn't even directly British territory, but belonged to the Hudson's Bay Company.
Besdies, at the time of the Purchase, there were only about 2,500 Russians in the whole territory, mostly trappers and merchants. Not any sort of sizeable military presence.
I imagine the British did not bother to take Alaska during the war because they had more than enough desolate, uesless Arctic territory on the continent - remember the Americans were reluctant to buy it 20 years later. The attack on Petropavlovsk was more to deny the Russians any Pacific access - a much bigger blow than occupying the remote, failed-as-a-mercantile-operation colony across the Bering.
It would have been interesting if the war had gone a little better for the British, and especially if they had managed to secure the Asia-Pacific coast of Russia if in the peace they had swapped Kamchatka for Alaska. A minor bargaining favour for the British, but they may take it as a concession so as to 'complete' their BNA territory...