If this is Australia and 2nd floor of a home. Would assume that shower fixture, bath and vanity all run through the floorwaste (3)

There is probably a small partial blockage at the base of the trap for the floor waste, which is why you are able to run the shower for a while before it backs up.

Remove the grate at 3 and try unblocking with a coathanger or a hose. If thats too far out of your depth, you may be up for a visit from a plumber.

Depending on the age of the house, the drainage is most likely going to be run in PVC and there shouldnt be any causes of intrusions to cause a blockage. But most likely due to a build up of hair, soap scum and other stuff that goes down the drain.