(By non-school leaver I'm assuming you mean people with a tertiary record.)

It depends on which university's undergraduate medicine degree. Here are some examples:

For Monash Univeristy, they only accept standard applicants with no tertiary record. If you miss out one year because of UMAT or ENTER/ATAR scores being too low, you could choose to take a gap year to try again. If you decide to start university, you are not eligible to apply anymore.

For The University of Adelaide, non-school leavers are accepted only within the uni. That is, you can transfer into medicine only if you're already a student there. There is a low intake though.

For the University of New South Wales, you may apply but if you've completed a full year of study at tertiary level, they will take your GPA with ENTER into account. UNSW: Can I gain admission after partly completing or completing another undergraduate degree?

Link to a table (see non-standard column). Take note that it hasn't been updated, since Melbourne doesn't offer undergraduate medicine anymore.

Also note that you are allowed to sit the UMAT as many times as you want.