My read...I assume the daughter is above 18 - as QF have some inconsistently applied guest rules for children under 18.

EDI-LHR-SIN-SYD in J
J lounges with a +1 (daughter) in EDI and LHR.
QR SIN lounge I doubt would be open at that time
BA Concorde Bar is only for F and GGL
I would go for QF J for the showers - if that is not applicable then I would go BA J - only because the boarding time of BA15 can be funky and can move very quickly from "go to gate" to "flight closing". Showers are not that great in BA J lounge (small and humid).
Note QF J and BA J are essentially next door so you could shower, grabs some food (food is better in QF J) and then head to BA J to get the boarding call.
Also note in SIN, if you are doing BA15 all the way, you do not have much time.

SYD-DOH in F with QR
QF F with daughter as +1 based on BA Gold card
I doubt daughter will get access to QR bar. They tend to be very by the book.

DOH-LHR-EDI in J.
J lounges (you cannot access F lounges based on status anyway)
Not sure if QR allow guests in Y in their lounges so you could be the generic Gold lounge?

KF