Payment is dependent on the school policy.
In my last school I was HoD and I advised the school to pay the teachers after they put a claim form in. This would lead to no issues with payment.
There are some teachers that were self-employed but they were also paid by the school but it was up to the individual to pay their own taxes.
You have done the right thing by sending your CV and generally departments will look at new teachers from about June onwards. Some schools employ directly from local council/ music services. I have always looked at different music teacher sites and located my teachers based on their distance so I know they have less travelling time to consider. I have made exceptions. Piano will always be a popular instrument but some schools may have a few teachers.
I dont teacher piano in schools as from my experience, parents are reluctant to have children out from lessons and payment is based on the work you do during term time so during the holidays you will be a little short naturally.