I have also done the LHR-SIN-SYD roundtrip in F with wife, with SIN legs on the A380 and SYD legs on 777. There are bassinet positions on all the planes in F (4F on A380 and 777) but on two legs they did not load enough cots, but thankfully there were empty First seats so my daughter got her own bed.
There is even a cot position on 787-900 (2F) but it is useless, since blocks out the whole seat, and my wife couldn't put her legs out.
A380 is not bad in F with a cot, but for 777 and 787-900, I have a clear preference for Club, rather than First, purely to (my wife's) personal discomfort... daughter is a bit of a mama's girl...