Nipping and mouthiness is definitely something that cattle dogs are known for. Biting is a really common problem in the breed as a result. I would suspect, just from knowing the breed (since there could be a lot of factors going on, I can't say for sure of course), that this is aggressive herding behavior rather than attempts to harm. Heelers are notoriously difficult to have around small children because they have extremely strong herding drive that kids tend to trigger, and they also have an aggressive herding style unlike some other herding breeds that may have more bite inhibition and a more "hands off" style. It doesn't take that hard of a bite to break skin and leave bruises, especially on a child. Heelers are also known for their "dog police" role--they tend to take it upon themselves to control and police the behavior of everyone in the house if not trained properly. It could also be that the dog is biting out of annoyance or as a correction for the kids, in its self-appointed role as police. Those are the two most common problems I've seen, rather than true aggression.
edit: sorry, my own red heeler was trying to get me to play and hit 'reply' for me.
I think you're right to insist on muzzling and not let kids around the dog. Even if it isn't aggressive behavior (in the sense of the dog actually trying to harm kids), this is a big problem and I'm sure very traumatic for the children. One of the biggest rules responsible owners follow with herding breeds is never to let them herd or nip people. Your sister has screwed up here by letting it escalate to this point.
I do think the trainer's statement is true. Some dogs don't like kids, and that can't really be trained out (although a combination of proper management and desensitization can get the dog to tolerate kids if they're only visiting). However, if this is a herding or policing issue, then the dog can definitely learn not to do it. However, IME it is extremely important that the dog is given another job. These drives tend to be very strong in heelers, and if they're not given a job, they'll find one for themselves--and I can guarantee it won't be one you like.
That's all I really have time to write at the moment as I've got a training to head off to, but I'll try to either post again later or PM you some more information. Just wanted to confirm that this isn't an uncommon problem in the breed (I did ACD rescue for years and many of our dogs came in because of issues like this), that there is a lot you can do to fix it although there are never any guarantees with animals, and that you're right to be taking it seriously.