The very fastest way is if you have an e-voucher, since you can wizz through things very quickly, bearing in mind one also has a 24 hour window if a big mistake is made. The thing that sometimes gets in the way is that the credit card check windows, which aren't controlled by BA, sometimes go down briefly in the middle of the night, so ideally you would have (e.g.) an Amex card and a Visa card on your profile, plus Paypal as a backup. Having 2 Visa cards only may not work so well.