Hahahah relatable.
On a serious note though, I had this issue up until recently when I took it into my own hands to fix as I get better customer support from the piece of dog shit outside of my office than I do with BT.
PSA : BT HOME HUBS ARE SHIT.
I went to Amazon and ordered myself a £40 Netgear hub. Didn't need a new modem.
Installed that & binned my BT homehub, renamed & configured everything to work with my BT Infinity line (which is really straightforward), and now I no longer have disconnection issues and my speeds are a lot higher. BT Broadband itself isn't bad, it's just their supplied home hubs, definitely something fishy going on there, perhaps a money-grabbing scheme similar to Apple's Iphones slowing down etc. So yeah, get yourself a netgear router and the issues should be resolved.
EDIT : Had the same issue in our office with a BT router, did the same thing (switched out for a Netgear router) and this fixed too.