BA do sometimes do so but rarely. Not "can't rely on it but I won't be surprised", not even "It happens sometimes", but "I can't remember this happening to me before anytime recently".
It seems to be mostly if they wish to impress someone, like a newly minted Gold member who looks like they shifted their allegiance recently. So it doesn't apply to most people.
Your chance of being able to get a cheap paid upgrade (either before departure via the website/app, or at the airport) is medium.
Your chance of free upgrade remains very low.