Tax residency and the visa you hold are often not related. You may hold a WHM visa and be considered a tax resident (although you would still be taxed at WHM rates unless you are from a NDA country - see below)
Tax rates – working holiday maker | Australian Taxation Office
From the Department of Home Affairs website, a bridging visa comes into effect when your WHM visa expires. From their website (check yourself as well)
"The bridging visa comes into effect when your current Substantive visa expires. You need to comply with the conditions of your current visa until the bridging visa comes into effect"
So I would assume, if your WHM visa hasn't expired then nothing changes with your employer.
When it does, provide them with a withholding declaration form.
Withholding declaration | Australian Taxation Office