On shared sites, its quite possible for EE to be working, but no slots available to BT Mobile, it depends on how they are allocated.
No Service can simply mean that all of the available BT Mobile slots have been used up, but EE could still have spare slots and would show a good signal.
EE mobile status is not really a good guide.
I could not find any reference to mobile cell status from the link provided. All I could see were for landline, broadband and BT TV, not mobile. Normally I get 3/4G but today my phone has been showing Edge for most of the day. The EE status shows they are upgrading some local cells, but I wondered if there was an equivalent from BT.
There's an app you can get; Ka i Bits, Network Signal Info Pro. There's a free version, or paid version.
It will show which cell you are connected to, and available cells. It just gives that bit more information, on what is going on.
From Keith's post it's just possible that you are shared with EE, and not separate.
Sometimes the 3g/4g signal will go away, revert to Edge on a different cell, and return later in the day.
It could be that cells are being upgraded.
Every so often, here, more cells appear on the map, and the signal improves, or at least it's more consistent.
Thanks for the info, Keith.
Have you any idea why I'm unable to use the BT Mobile troubleshooter page? I log into my BT account, and go to the troubleshooter page. It pre-recognises my mobile number (towards the top of the page), but further down is a "picture" which is like a screen, within which there is the statement "Image not available". Therefore I can't go any further - my (correct) mobile number can't be confirmed.