This answer is not really an answer, more of an excuse to prevaricate about arcane baggage rules, but I set out some of the detail around this question.
IATA Resolution 302.
1 April 2011.
RESOLVED that, unless otherwise agreed the following baggage provisions selection process should apply for interline journeys ...
There is an IATA rule governing baggage fees on interline journeys, IATA Resolution 302. The Resolution is not freely available but several airlines and at least one government have been kind enough to publish it.
The Resolution sets out the minimum that an airline is expected to provide. They can certainly do more.
Essentially, for each baggage journey, the most significant carrier ("MSC") determines the luggage allowance, together with any fees for exceeding the allowance, for the whole of the baggage journey. This makes sense because the most significant carrier is the one who has to do the most work to transport the bag. A baggage journey, in this context, is from check in to pick up, so for all connecting flights, but not for flights taken at a later date, even on the same ticket. Other baggage journeys on the same ticket may, in principle, have different allowances; so the return leg could differ from the outbound leg.
The most significant carrier is essentially the one flying the longest leg. More specifically, the MSC is the first carrier to cross from one IATA Area to another, or if that doesn't happen, the first carrier to cross from one IATA Sub-area to another, or if that doesn't happen, the first carrier to operate an international leg, or if that doesn't happen, the first carrier.
In your case, Finnair crosses from IATA Area 2 (Europe/Middle East/Africa/Russia west of the Urals) to IATA Area 3 (Asia/Australasia/Russia east of the Urals), so it determines the rules for BA to follow.
Local laws may overrule airlines' policies.
If your journey had begun or ended in the United States, the above rule is superseded by a federal law that requires that the first carrier's fees and allowance to apply throughout the ticket; and further more requires that the codeshare carrier determines the rules, not the operating carrier.
In practice, when anyone books an itinerary including points in the United States, the ticket will simply apply the most generous allowance that applied on any leg throughout the entire journey. [I suspect this is because applying the first carrier's allowance, when you told the pax he would get a better allowance later, would be legally unstable territory. (In my case I usually end up with four 32 kg bags on all tickets touching the USA, which I do find faintly ridiculous.)]
Other countries apply other laws, such as Brazil who requires everyone to get at least two pieces for free.
Some carriers always apply the most generous allowance from any leg throughout the whole ticket.
Back to the question at hand, British Airways will indeed apply the most generous allowance accruing on any one leg throughout the whole ticket (even if it does not touch the United States). But that only applies if BA is the ticketing carrier. Although it is possible that BA plated your ticket, it sounds like Finnair would be likely to be the ones who "printed" your e-ticket. In this case, you will have to defer to their policy: since the luggage allowance per leg is either written or stored in the ticket, if Finnair issued the ticket, they get to decide what the luggage allowance is.
Essentially when BA check you in, they will inspect your e-ticket. The check in agent does not need to do any of the calculations we have discussed here to determine how many bags you get. The right answer is already either encoded on the e-ticket on printed on in the "PC" (pieces) column on the paper ticket. (The agent may give you an extra bag in accordance with your airline status; this is not always printed on the ticket already.)
Your upgrade happened after the ticket was issued.
There is a further complication in that if your ticket was not re-issued or revalidated, then only Finnair know that you have been upgraded. In this case the ticket may still reflect the old allowance.