I have made a formal complaint around issues to do with transferring from BT to EE, so I sympathise. Still have no working landline over a month after the switchover, and at this point, I doubt that EE are able to provide a service or get my old number back... But fingers crossed for you.
As mentioned the BT router should work with the EE broadband connection. Whether it requires resetting or any setting change, I don't know. If you ever get through to EE (if you can't use 150 then you can use 0330 123 1105 from another network), I have found using the technical support option rather than general enquiries seems to get through to people who are a bit more helpful.
Regarding your mobile phone switchover, did you get a new SIM card from EE? If so is it the EE SIM which is not working, or your old BT SIM? If your experience is anything like mine, you will receive a new EE SIM which will need to be swapped over in your phone. It will have a new random phone number attached to it, but it should at least work. When you transferred initially you should have received a text message from BT containing a PAC code which will be needed to transfer your old BT number across to EE. In my case the number transfer took around 3 weeks, and took a few phone calls to chase them along, but I understand it doesn't usually take this long.
