First, I want to be sure I understand what you are saying about other computers in other locations: you have tried exactly the same situation there as "at home", right? I ask because WordPress behaviour when you are logged on versus being logged off, can be quite different even on the public web site content.
If that is true, I can only see two possible solutions:
- Replace the Modem or Gateway or whatever device it is that contains the Modem, with a different model, preferably by a different manufacturer
- Change ISPs. Hopefully BT doesn't have a monopoly
Based on what you have described, those are the only two possibilities that I can think of. I had a similar problem with an IPTV box at a relative's house. To absolutely positively be sure that I had properly nailed down the problem, I ran both ISPs for a few days. One worked. One didn't. Two years later, the IPTV box still works. For 6 months before that, there was no service for several hours each day. The old ISP admitted to blocking certain types of video streaming.
Not that WordPress and IPTV have anything in common. But ISPs are sometimes woefully unable to resolve "weird" problems.