As I see it, your mate's ex signed the contract, with the phone company, for the provision of telephone services to that address. So, even if your mate phones his family in Australia every day, she is still responsible for paying the service provider.
However, she could argue that because it's 'her' phone service, he's no longer entitled to use it (or certainly not without paying for the calls). She could then bill him for the cost of all calls made since she left. (She can't bill him for such things as the line rental, because those costs would have occurred even if your mate had pulled the phone's plug out of its socket and never used it. It's her fault that her contract with the phone company was never terminated).
Concerning who should contact the phone company, it would seem to make sense for your mate's ex to contact them, to prevent her receiving further bills for a service she's no longer using. However, your mate needs to be aware that, if she does so, he might suddenly find that his phone line has gone dead and that he's required to pay a connection fee as a new customer. So, it might be simplest if he just asked the phone company to change the name on the bill.
Chris