Melbourne - Adelaide (Qantas Economy) 2hr transit
Lounge Access - Qantas Domestic Business Lounge
You won't need to use a lounge pass for access in Melbourne. Access granted based on later ADL-KUL boarding pass.
Oneworld rules allow those travelling on long haul business class access to lounges for connecting flights. Simply show your onward ADL-KUL boarding pass or itinerary for access.
"First and Business Class customers connecting on the same day of travel, or before 6am the following day, can access the lounge when travelling between an international long haul (a oneworld international long haul flight is defined as an international flight marketed and operated by any oneworld carrier with a scheduled flight time longer than 5 hours) and an international short haul or domestic flight (and vice-versa)."
Adelaide - Kuala Lumpur (Malaysian Business)
Lounge Access - Qantas Club Adelaide
Access granted due to the fact that you're travelling in Business Class. Just show the MH boarding pass for access.
Kuala Lumpur - Dubai (Emirates Business) 1hr 45min transit
Lounge Access - Plaza Premium Lounge
Access granted due to the fact that you're travelling in Business Class. Emirates will likely issue a voucher or pass at check in to show the lounge otherwise just show your business class EK boarding pass.
Dubai - London Gatwick (Emirates Business)
Lounge Access - Emirates Business Class
Access granted due to the fact that you're travelling in Business Class. Just show the EK boarding pass for access. Note that emirates has 3 business class lounges in Dubai. You don't have a long time so pick the one closest to the gate.
Rome - London Heathrow (British AW Economy) 1hr 45min transit
Lounge Access - British Airways Lounge Rome
Same as the first flight, access will be gratned as FCO-LHR is short haul but onward is long haul business class, show both the FCO-LHR and LHR-DXB boarding pass.
London Heathrow - Dubai (British AW Business)
Lounge Access - British Airways Business Class Lounges London
Access granted due to the fact that you're travelling in Business Class. Just show the BA boarding pass for access. British Airways has 3 business class lounges in T5. Two are in T5A and the final one is in T5B (often quieter) but only go here if the onward flight is in concourse B
Dubai - Singapore (Emirates Economy)
No Lounge Access Here
There are numerous paid lounges you can purchase access to otherwise the Emirates lounge also allows paid entry for up to 4 hours at ~AU$276 (current exchange rate) to enter their business lounge or ~AU$473 to enter the first class lounge.
Singapore to Melbourne (Jetstar Business)
Possible Lounge Access - Qantas Business Lounge T1
Jetstar Business class does not include lounge access (unless you have purchased the max bundle). Now it is possible to use one of your invitations here however the big thing to note is the change of terminal required. The Qantas lounge is located in Terminal 1 whereas all Jetstar flights depart from T4 which requires a shuttle bus.
Terminal 4 is not connected to T1/2/3 at all and requires a shuttle bus both landside and airside. Whilst technically possible, you'd need to check in at T4 and then head airside before catching a shuttle bus to T1, then after visiting the Qantas Lounge take the bus back to T1.
However there is one exception to above, From 12pm until 8pm daily, Jetstar offer an alternative check in area at Jewel Early Check in (connected to T1). The catch is you must use this facility more than 3 hours prior to departure. So an example is you can show up and check in at Jewel 4 hours prior to departure, head air side into T1 and use the Qantas lounge followed by the shuttle to T4, although ideally you'll want to leave the Qantas lounge at least 1 hour prior to departure.
The Qantas lounge at T1 opens at 2:40pm and Jetstar offers early check in from 12pm so there is nothing stopping you from spending several hours in the Qantas lounge.
