Unfortunately BA does not extend the BA baggage prices to codeshare flights. If you have a transit in SIN, then no additional charge by QF, but if you have a stopover at SIN then QF checkin will charge you the QF fee for a bicycle, which is quite expensive compared to BA (about GBP 200 for 15kg, for example). For the return trip you'll have to pay QF and also meet QF's stricter bicycle packaging requirements.