I would be surprised if this was the case. One obvious thing you can do is take yourself off to a coffee shop for an hour or so with the laptop or an App filled Smartphone and see what happens if you rebook yourself, e.g. Lufthansa and another hotel, for near immediate travel. You then compare that with what you paid BA. You then ring up BA Holidays in this case and tell them your intention to cancel for a full refund and pursue for "loss of enjoyment" and/or "loss of value" - BA Holidays have a dedicated team who may be able to fix things. If it was mainline BA I doubt they would be able to do much, but log the agent's name and that makes the cancellation process easier. If no dice with BA ring your insurers for a similar conversation. You can then make a decision as to what to do from a position of strength. Short notice bookings on empty flights / hotels can be cheaper than those planned in advance.

Not applicable in your case, but let's say you had travelled to LHR from UK domestic or the rest of Europe, and then hit this situation: after 5 hours you can ask to be returned free of charge back to your start point AND get a full refund. Airlines don't mention this since they hope people don't do it.