I think it depends how much time they reasonably left, and the cause. This clause is often called the "flat tyre clause"... If it's a bit of traffic that could have been foreseen, and you allowed 15 mins for error, BA could say this wasn't really beyond your control. If it's an accident or a stuck train or something, I think BA would be more sympathetic.

To save people looking it up, here's the clause:

I've tested this several times, but happened to make each flight with minutes to spare. Each time I called from the train to let them know - on BA LHR-ZRH and TLV-LHR, on IB JFK-MAD, and on AA JFK-LHR. In each case I was on the train to the airport having allowed an extra 45 mins before conformance for any potential delays (GCH/usually HBO/Global Entry/OLCI/BA app/Israeli/British so this is really plenty, before the naysayers kick in!), and something went awry between stations with no other options available (the IB flight there was a pile-up on the freeway).

On the BA and IB flights, I was told not to worry, they would keep an eye out and try to expedite me once I got to the airport and make arrangements if not. LHR-ZRH they actually someone waiting for me with a boarding pass and bag tags by the front desk of the F check-in area, so I could just hit security straight away.

On the AA flight, they were completely unsympathetic, which was onerous as it was the last AA flight of the day and they refused to pro-actively move me to the half-empty last BA flight going 30 mins later until I officially showed up too late, which I would have missed in turn due to the terminal change. I bribed the taxi driver who picked me up to drive like a nutter, and was the last person to make the original AA flight, as they were calling for me in the terminal.