Alaska is a big strategic nothing regard a threat to Russia. If it were, Russia wouldn't have sold it, and if Russia was selling, Britain would have bought it. No one, No way, no how, is using Alaska as a threat to Russia.
Russia, quite correctly, knew it was not vital to Russian interests, industry, military, or expansion, and it was over extending themselves to maintain a road to nowhere. Thus, it was sell it to anyone who would give them a nickel for it. The US wanted it because it furthered their interests. As someone above said, the Seward's Folly moniker was a political slogan. the US, overall, wanted it. Britain was busy making oodles in India, and didn't see Canada as much of a area of desire, let alone spending money to expand it.
Overall, had Britain obtained it, nothing much changes. the mineral riches go to a different country. that's about it.