@Dani1981 - as @Sister and @"Annie C" have said - welcome to the site - and NEVER feel guilty about putting questions up - if we can help you & your mum have an 'easier path' at this early stage of diagnosis & surgery - that will be a bonus.
Hopefully we'll be able to help lessen the impact that this shitty disease has on both you and your Mum, as you help her thru it. If you are going with her to appointments, consider recording them on your phone to 'go over' at a later date, if required, as it is very difficult to take it all in on the actual day.
Re Private or Public - there can often quite large 'out of pocket' expenses with private surgery (I'm up to about $6000 now with my private surgery & other gaps.) So make sure you ask your surgeon - what the gaps are if you go private & also if he works in the public system (most do.) It may mean a small 'wait' of a week or so for the surgery .... but this should not affect your Mum's outcome at all (except that you can use the money that is saved - to take your Mum off on a great holiday at the end of her active treatment!) I went public for my radiation & couldn't fault it.
Well done for stepping up to the plate & helping your mum thru this - I am so sorry to hear of your Dad's passing. :(
Take care xx