I've been transferred a few times with BA to the next available flight with non-flex and even Avios tickets. Either free of charge or with a £60 fee, both at Heathrow and at outstations, so it doesn't seem to be a blanket rule. Though on all occasions flights have been very quiet - can't imagine they'd be so generous at peak times.
I like Easyjet's rule - arrive within 2hrs after departure and they'll "rescue" your ticket for about 50 quid confirmed onto the next available flight. Not exactly cheap but at least you aren't relying on discrection. Combined with their free space-available move to an earlier flight home they are usually my carrier of choice on my short european hops these days