I have been having a very similar problem for a few weeks now, also with iTunes UK. I say similar, because it clearly is related to what people are describing here, but there are a couple of significant differences.
In my case I seem to have a number of "phantom" available downloads. That is I am absolutely certain that there is nothing that needs to be downloaded, but iTunes or the iTunes store (or both) think there is.
I can interact with the store just fine: I can sign in, sign out, browse the store and make purchases or rent movies and they download just fine (eventually, read on for an important detail).
The problem is that iTunes seems to be convined that I have downloads available. I have "automatically check for downloads" enabled in my preferences. So when I start up iTunes it seems to think that there is stuff waiting to download, and then proceeds to prompt me for my iTunes password up to 3 or 4 times. Then iTunes will become unresponsive while it attempts to connect to the store. Eventually these attempts time out, with the error dialog "Unable to check for downloads. Network connection was lost". It seems like I will have one of these for each login dialog I was presented with earlier, and until all of them have timed out I can't really do anything else. However, once they have timed out I can resume interacting with store and download purchases.
For the last couple of weeks iTunes said that I had 2 available downloads -- as I said I am convinced I do not. This morning an album I pre-ordered became available. The album has 11 tracks. Now iTunes says I have 13 available downloads, BUT -- and this is significant -- the album has downloaded anyway and I have been listening to it today. That is why I said at the beginning that these in my case seem to be "phantom downloads".
This is very frustrating because although I can still use iTunes the issue with it attempting to download what it thinks are available downloads, then prompting me over and over again for my password, which then eventually times out, is happening fairly frequently, not only when I start iTunes.