As a citizen of a Commonwealth country resident in the UK, I am eligible to vote in UK elections. And I've just sent off a letter to the Oxfordshire Council registering as a voter so I can do that.
My first stay in the UK was as a child in 1979, which saw a change of government, and I was also in the country for the 1997 election campaign, which saw another. The imminent 2010 election looks like being an interesting one too. We're in East Oxford, which is held by Labour but under threat from the Liberal Democrats.
I can still vote in Australian elections, as well. (And the NSW election may see a Green challenge to the Labor hold on Balmain.) One of the things I did before leaving was to register with the Australian Electoral Commission as an "eligible overseas elector" and a "general postal voter". They sent a letter confirming this to my UK address and, since I had no letter from an employer like C, this turned out to be the only document I had which had that address on it - which I needed to open a bank account!
So if you're an Australian moving overseas who may not have documentation of your new address - perhaps because you are staying with a friend temporarily - I recommend registration as an overseas postal voter. The Electoral Commission letter looks impressive and has both your new address and your old Australian address on it, so it could be used in conjunction with Australian documentation referencing the latter.