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One really solid leader could speed that process up a lot. Similar to how everybody was making fun of Bush and the US, but then Obama came along and despite his policies sometimes being rather disagreeable as well, everyone was suddenly fine with it. Obviously this disaster is much worse than having someone like Bush in power, so it will take longer to recover from, but it will also really depend on who's prime minister next.
The problem is that during the confidence vote, TV was showing potential Tory hopefuls. Not a single name that didn't make me feel queasy... So at least on that side, there's not much hope of an actually 'good' leader.
To paraphrase a book title, this whole thing is a bonfire of the credibilities.
It will take 20+ years to restore the UK's diplomatic reputation