An upgrade is basically a change to your booking so you normally need to speak to whatever reservation/sales team you booked your ticket from in the first place, be it 1) the Ba sales/reservation team (cash flight only booking direct from Ba), 2) your ta/ota (if you booked from a third party eg orbitz, Expedia etc), 3) the Ba holiday team (if you booked a Ba holiday package).

Those will not "touch" each other's booking, ie if you call BA but you booked through Expedia, BA will not touch the booking, and the same if you booked a BA Holiday booking so it's important to go back to the source of original purchase. As mentioned, if you booked a BA flight only online, then the best number is whichever one you normally use to speak to them based on location and status as there is no separate service for upgrades.