Well, after spending six hours transfering files and making a copy of the board's database to upload to the new server, I am now unable to access the control panel to complete the last few file uploads.
I have to wait for their office hours to get an answer, but I'm not too impressed about this failure of service on the first day.
I'll try again tomorrow night. (I can't do it during the day, because the board has to be turned off during the copying of the database and uploading it, which is over an hour to do, and a fresh copy needs to be made with each attempt).
I'm off to sleep now. Have to get up in 4 or 5 hours time so I can verbally abuse the new server host.