I am an Aussie!, and know what you mean about not being able to understand some accents. When we first got here, nearly 9 years ago, DH almost got a job in Barnsley. We went up there to investigate, and went to a sandwich bar for lunch one day. We couldn't understand them, and they couldn't understand us!
A bit more seriously, I don't think you need to not have an accent to be a speech therapist. We took DS1 to one, and it wasn't just about the therapist's accent (she was also Australian but had spent a lot of time in Canada so had a very strong Canadian accent) but more about how we interact with him.
The accent didn't come into it all (well, not at his age anyway (3)).
So, I don't see why your SIL couldn't do it, if her accent is the only thing holding her back...
A lot of overseas students train to be SLTs in the uk, and many of them have quite strong accents. Speech therapy is often more about communication than speech, so as long as the accent isn't impossible for others to understand, it shouldn't be a problem.
I've worked with Greek, Malaysian, South African, Australian and New Zealand SLTs, and am Scottish myself. Everyone has an accent of some kind.
Best of luck to your sil - where is she hoping to train?