I'd like to add to what Anton has said. Namely, we are planning to add a simpler version of the chat right into the Full control / View window. The old Chat mode will still be there in case you need more chat features. It's only that the simple chat window/pane (at the right hand side of the screen) will allow you to quickly exchange messages with the remote user during remote support without switching to a separate chat window.
Are there any hosted GMS servers in the UK? (other than hosting our own, of course). If not, any plans to open other public GMS servers?
We are planning on deployment European servers very soon.
If we were to change our customers to our own GMS, can we do this remotely?
You change the mediation server address in the "Internet ID settings -> Advanced" on the Host. Yes, during a session you can remotely change the server the Host is pointing at. However, the remote customer will have to restart the Host (Stop Host then Start Host) so that the Host applies the new setting. Or you can simply reboot the remote PC, or reboot the Host service using a .bat script.