In that setting you can go direct to an Eye surgeon (a medical doctor, not an optometrist where a glasses shop is) or to your GP who can send a referral to your local hospital.
Why not an optometrist? Well, your child has to have this problem solved as quickly as possible, some optometrists are good at managing children with difficult problems, others aren’t, but you won’t know who is who. Whoever you see they must do a full dilated pupil examination, and promptly organise treatment, including patching, so that your child’s vision begins to respond. If they don’t seem to be getting better ask for referral to an Eye specialist (doctor).
An Eye specialist (doctor) will not give glasses unless your child really needs them. But many optometry shops will always give glasses regardless – you have every right to ask for a 2nd opinion if you feel glasses are being sold to you without a significant problem being present.
If you qualify for the enable subsidy then a full eye check from either an optometrist or Eye specialist should be provided free of cost.