Start by contacting your airline after booking to see if there are any paid upgrades available. If you're out of luck there, you can try again at the check-in desk. If it's another no, you still have a chance.
Can I pay for an upgrade on BA?
To check if an upgrade is available on your flight, please speak to a member of British Airways staff at the airport who can advise you of the current availability and price. Customers who purchase an upgrade can benefit from the full entitlements of the upgraded cabin, including increased baggage allowances.