One World Elite Baggage Allowance
 

As someone who has the misfortune of having to often check baggage due to a back problem, I've noted that Star Alliance offers a substantial additional free allowance to Gold members of their FF programs. I've not seen anything similar for OneWorld as a whole.

Does anyone know whether any individual carriers within OneWorld offer additional allowance for Elite members of the alliance (not their own Elites)? Thanks.

Additional baggage allowance is not an OW elite benefit. Individual airlines do provide additional allowance for their own elites (eg CX, AY, QF) but it only applies when flying with that particular airline.
The only possible 'alliance wide' unofficial elite benefit is that OW elites are generally not affected by the recent wave of reduction in TATL baggage allowance. (eg AA, BA)

If on the one PNR, BA will provide the most generous allowance for the passeger on their itinerary.

If on the one itinerary (can be more than one PNR) for international travel, Qantas will allow up to two pieces at 32/23Kg (depending upon travel class) if the itinerary includes travel via the pacific.

If on the one itinerary (can be more than one PNR) American will allow checked baggage greater than their standard or not charge fees in some cases.

But, no, other than BA there no alliance check baggage benefit. With BA this is only applicable if your PNR includes travel on you 'home' program's carrier where you get that extra allowance.