A double-cabin upgrade, paid or otherwise, is a
very rare event on BA. It should not be possible except in cases of disruption. The official BA policy is a maximum upgrade of one cabin.
From World Traveller, you may only upgrade to World Traveller Plus - except for Short-Haul or certain Medium Haul routes where WTP does not exist, in which case you may upgrade to Club Europe/Club World).
Where WTP is available, the cabins are small (smaller than Club World) and are often full as they are very popular with business travellers (who will be hoping to score an upgrade to Club World - ha!).
Paid upgrades may be available in advance via the Manage My Booking function on ba.com, at check-in online (including via the app) and are also available for purchase at the airport, if you ask.
If you book your ticket with a travel agent, upgrading online is difficult, if not impossible, because (AIUI) the travel agent "owns" the booking and BA cannot modify it until it is released to airport control at check in time. In theory, the travel agent can do the upgrade but in practice I have never been successful in getting this to work, but maybe a different travel agent might have more luck, knowledge or diligence.
Online upgrades are also very hard to get if you have a connecting flight on your ticket - for some reason BA want to upgrade every sector, even if all you want is just a one-sector upgrade. If this is the case, an airport upgrade is your best bet - though you could try calling BA.
I should also mention that if you have a flight with any other carrier (even within OneWorld) on the same ticket, BA will not permit any upgrade.
Even if seats are empty, upgrades may not be available - or may not be available
to you, depending on how overbooked the cabin you booked is, your ticket/booking class (i.e. full fare Y, or massively discounted Q) and/or frequent flyer status - or perhaps more correctly, your commercial value to the airline.
The moral of the story is that BA would sort of prefer to avoid the whole unpleasantness of upgrading people and would much prefer you to book and pay for the cabin you actually want to fly in. Unlike some carriers, they will permit a premium cabin/seats to remain empty, rather than fill those empty seats with upgrades - even paid ones.
My advice - if you're hoping for an upgrade of any kind, BA are probably the worst choice of airline on the planet.
