I imagine that one of the immediate effects will be that LEA staff will simply become freelance and sell their services to MATs at consultancy rates which will be way more than what schools are presently paying for them.
This has already happened in many LAs. Where I am, the LA no longer delivers schools H&S, HR, Equality or payroll services to name a few. The staff doing those jobs in the LA were made redundant and the schools have to buy in those services from the private sector.
Several of the former LA staff set up in business and are now contracted by the school.
Interestingly though, having spoken to several school business managers, they say that the cost of these services is less now they are buying direct than when their budgets were top-sliced by the LA to deliver direct.
And of course, many MATs have their own services, or commission them - so there are economies of scale for many of those services and added value - so, counselling services for staff as part of a HR package for instance, which can't be offered through the LA.