I once tried to upgrade a paid J ticket to F in HKG for a HKG-JFK flight. It was a corporate J fare (C class IIRC), and after an hour or so (while I waited in the lounge) they came back and said they couldn't figure out my fare basis and thus couldn't calculate how much it would cost to upgrade my fare, so I was out of luck.

Given that CX contracts to BA staff in JFK, plus the fact that the fare basis on a reward ticket is ZERO, I'd say the chances of a cash upgrade are less than zero.