I do not believe it is for "throughout your itinerary". My understanding is that it is based on the bag journey, i.e. from when the bag is checked in to pick-up. In other words, a break in the journey of >24 hours or an short checking them will "terminate" the bag journey and therefore the allowance of the most significant carrier for the next bag journey will be applicable.

I may have got the wrong end of the stick but the "whole itinerary rule" applies only for journeys involving (or originating or terminating in, rather than via) the US and that's due to US DOT ruling, and normally the IATA res 302 applies only to each bag journey.

So, my understanding is that if the OP is flying LHR>HKG (BA CW) - STOP - / HKG>SIN (CX PE) - STOP -/ SIN>LHR (BA CW), then the allowances would be 96 kg (32 x 3) / 25 kg / 96 kg (32 x 3).

However, if it's LHR> (BA CW) HKG > (CX PE) SIN with no stopover in HKG, then BA allowances will apply. If it's HKG > (CX PE) SIN > (BA CW) LHR with no stopover in SIN, then BA allowances will also apply.

From what the OP posted initially I was under the impression that there was a stopover at HKG and SIN or at least in HKG but if there is no stopover in HKG then BA's allowances can apply to the whole journey (certainly if the whole journey was in J).

One thing that sticks a little here is that the OP is travelling in different classes and the CX website is very specific about mixed class journeys in terms of baggage allowance of each sector being applicable, so I would personally not count on CW allowances being applicable when flying ex-HKG (in any case without any argument) even if there had been no stopover at SIN.

If there is no stopover at HKG, and if BA tags them all the way through to SIN (that is, if they can do so without CX charging excess baggage fees on the basis of mixed class specification stated on their website) then HKG-SIN leg should be fine with BA allowances though.

BA agent was correct in that had there been no stopover in SIN, BA's allowances would have applied to HKG-SIN - that's due to MSC (BA would be the MSC as HKG-SIN does not cross any area or subarea and therefore the first sector that crossed the area or subarea is on BA, rendering BA the MSC for that part of the journey).