I am surprised that they knew that the SSID had been changed. Did they find out through their remote router management?
Ooh, I've missed that question, sorry. Yes, EE support are notorious for connecting to your router via remote management whenever you call them, and not asking for a permission. Here's one example of a conversation I had:
Me: my BrightBox range is too large, is there a way to reduce transmit power?
Sup: I am having trouble connecting to your router, could you power it off and on?
Me: I don't want you to connect just now, just tell me how it's done.
Sup: It is not possible.
End.
(it's a bit of an off topic, sorry)