Firstly, thanks Mwenenzi for the helpful info that I could pass on to my mum!

for anyone interested, the QF82 flight ended up 45mins delayed but luckily the transfer desk at SYD sorted out her new boarding pass and her luggage, so she also made it in time for QF161/BA7425.
If she had to exit to landside and re-enter again, I doubt she would've made it

downside, her luggage didn't make it to Wellington but we heard this was also true for a few other passengers.

Still perplexed with the handling in SIN, considering its all OW flights in the end, and she was always due to be on the BA7425/QF161 flight to begin with.
Now she just has to wait for her bag and "Wellington on a good day"