You know I think this is something you could have found out for yourself, but since it's 0640 and I have nothing better to do I'll just tell you.
For UMAT, ACER places no limits on how many times you sit the exam, as long as you keep paying and are at least a year 12 student or beyond. Some students find that their performance doesn't change from year to year, others see small improvement, and yet others (with a little practice) see leaps in score - so repeating the UMAT can definitely be worthwhile.
Of the undergraduate schools in Australia, two are open to school leavers only and the rest will consider NRSLs within certain limits.
Monash (Vic) and Adelaide (SA) will only accept applications from school leavers, or students who have graduated and taken a year off, but NOT enrolled in any tertiary study. Adelaide does allow applications to transfer from students of other courses within the University of Adelaide, but NOT Dentistry. Monash has no such concessions.
UNSW (NSW), Melbourne (Vic) and Tasmania (Tas) will consider applicants based on a combination of high school and university results. UNSW measures academic performance in terms of UAI (50%) and GPA (50%) regardless of time spent at uni. This score is then considered as an equal component with Interview score and UMAT to determine selection ranks. Melbourne expects students to have an ENTER of at least 97, and will also consider university marks. Admission into the undergraduate stream will only be accepted from students who have completed less than 50% of their course - for single degrees this is three semesters, or one and a half years. Tasmania will consider applications from university students with a distinction or greater average, and for those with less than three semesters at university the TER must be greater than 97.
Newcastle (NSW) and UWA (WA) both consider NRSL applicants based entirely on their academic performance, with no consideration of school results (UAI). Newcastle requires a "close to credit average" performance, anyone reaching or exceeding this requirement is eligible for consideration. Selection is determined by interview performance, with ties broken by UMAT score. UWA has an elaborate TPS calculation system (best described as a nationally standardised GPA scale) provided by GAMSA. The minimum TPS for consideration is 5.5 (in the Sydney universities, this is a little above credit) and your ranking on the selection list is determined by an equal share of TPS, UMAT and Interview.
JCU (Qld) doesn't use the UMAT, and they rarely accept applications from non-rural and/or non-Queensland applicants. To be honest I don't know much about them and don't care either. You can try finding more information on their site, but I wouldn't bother.