If you are travelling on different airlines, under multiple bookings, then this is referred to as "interlining", rather than connecting.
Almost all major airlines have an interline agreement with each other. This allows baggage to be transferred from one flight to another, regardless of whether the ticket is on a single booking, or whether the airlines are part of the same alliance. The notable exceptions are low cost airlines, such as Ryanair, Easyjet etc (although some of these are starting to seek interline agreements).
Often when you are interlining, one airline may not be able to check you in to your next flight, but they will still be able to transfer your bag over to the next airline. This is due to the separate reservation systems used, and lack of tie-ins between them.
In such a case, they will check you in as far as they can, you then need to visit a transfer desk at the airport, who will check you in for the rest of your flights. You don't need to clear immigration/customs or collect your bag, it will be transferred for you.
This happened to me when flying Air France from Aberdeen to Bali, connecting to Qantas in Singapore. Air France check-in at Aberdeen checked me in as far as Singapore, but tagged my bag for Bali. At Singapore, I just needed to visit the transit desk, and was issued my remaining boarding pass.
However, as you are transferring from one OneWorld carrier to another, they will almost certainly be able to not only take your bag, but also check you in to the next flight.
Enjoy flying Qatar and Perth! I flew them to Perth exactly a year ago, although from Edinburgh. Also, be aware that you will need to change terminals in Heathrow.