Thanks for the info everyone.
They seem to have a new security cert as of last Friday 26/03/2010 which runs until 26/03/2011, and my IE doesn't recognise it as valid. It opens without an error, and with all the relevant padlocks, in Firefox.
I thought running Windows update might get a new list of valid cert suppliers or something for my IE browser, but it hasn't helped. Any other suggestions? (My date/time are fine.)