I don't want to deal with eBay at any time; once I get to the UK, I
> don't want to have to "drop by a Carphone Warehouse" (I want to head
> straight to my hotel and crash!); and so on.
If price is no object, there are companies that will sell you a GSM phone with European SIMs and ship them to you in the US, but it will cost you about three times the cost of doing it the normal way.
Buying a prepaid phone or a SIM in Europe is not a big deal. I was on my way from North America to Ireland, changing planes in London two days ago. My old SIM had expired, and I wanted one that could roam into Ireland, so while I was out having lunch with a friend, I stopped into one if the dozen mobile phone stores we walked past and got a T-Mobile prepaid SIM for 11 pounds, with the entire transaction from walking into the store to having the phone active taking about five minutes.
Any mobile phone store will sell you a prepaid phone with a SIM, too. It's not a cheap as ebay, but it's cheaper than buying a euro phone by mail order here and you can see what you're getting.
R's,
John