Upgrades using Avios come out of the same bucket as full redemptions. So you can easily see if that is the issue by making a dummy booking as if you were trying to get an F ticket only with Avios. If there is availability for a dummy booking then ring up BA to get your CW ticket upgraded. If not you may get a Pro-active upgrade if you press the cash upgrade button which may be on the right of the MMB screen.
Yes, if all else fails, you may get a cost effective Airport Upgrade Promotion at the airport before departure.