Here's how it worked for me recently. Flying Club on BA Long Haul. Silver card. Asked for upgrade at check-in - no deal. Went direct to the gate and tried again, had to sweat a bit but eventually got it just before departure. Was advised by gate boss that key criterion is price paid for ticket.
Conclusion: best if you have a pricey ticket, maybe booked at short notice. Don't bother at check-in, decisions are made at the gate. Grovel as much your pride will allow!
Spent next 12 hours in excellent First Class - well done BA!