So is this self transfer possible?

Soooo I may have been a bit silly but I booked a last minute trip from Melbourne to Vietnam for two weeks just before my flight from Melbourne to London(layover in Shanghai).

I was on the phone to Trip and they said to me that the best flight for price would be a specific one and it was still direct. HOWEVER, it arrives at 9:55am at the same terminal I need to catch the flight to London for 12:30pm. They said I would only need two hours to transfer.

I’m not sure how it will work if I need to go through immigration, I have a checked bag and know I need to check in and out. I have a passport that gets me through the fast track of security with facial scanning ect and to be honest I’ve never taken more than an hr and half checking ect but want other peoples advice specifically for Melbourne airport. I’m at UK citizen and have WHV that expires end of May for Australia. Do I need to print out my documents?

I’m going from terminal 2 of Melbourne arriving and departing. Have one checked bag(maybe I just don’t use the checked bag to avoid going through that process?)

Vietnam to Melbourne 6/05/26 arriving 9:55am 7/05/2025

Melbourne to London 07/05/2026 departing 12:30pm 07/05/2025

Both terminal 2.

Thank you in advance!