We've used HSBC for several years, specifically because we knew we would be moving from the US to the UK. Generally speaking, the service has been better in the states. We've had two US Premier bankers in six years, and here in the UK we've gone through about one every six-eight months.
The primary thing we went with them for was, as other folks mentioned, continuity of credit facilities and easily transferrable funds between countries. The global view thing is MUCH better than it used to be, and it seems to work pretty well now. The only hassle is that you still do have multiple logins, one for each country--you just don't have to remember or use them now!
When we got to the UK and got ready to get a mortgage, they turned us down. So our Premier relationship didn't mean much, apparently. But otherwise we've had pretty decent service. When we went to set up a Euro account in France last year, it was fairly easy and has gone off without a hitch.
Unlike a couple of other folks have mentioned, though, we've not had to "re-qualify" in each locale.