As I understand it, a Statement would not be enforcable in a private school, as the school is accountable to its board of governors, not the state.
If the school is suitable and supportive, they might be willing to provide the support specified in the statement through their own funds. But they are under no obligation to.
As pp have said, it is unlikely they will provide any external support (SALT, OT etc) specified on the statement. But if it's a TA they'll probably be glad of additional help (like any school would be).
The only other option is to have a private school specified in a statement, and have the LA pay for it. This is pretty hard to achieve - there would have to be a pretty solid argument that the school is the only one that can meet the child's needs.
We have just had to make exactly this decision: ds (asd) starts reception in September, and had the choice of a fantastic prep or the good state school of our choice. We reluctantly went with state school in the end, partly to save money, but also (more importantly) although the private school could not have been more lovely and welcoming, ds's statement is for 25 hours per week, and they would not commit to providing this in full (and we would probably have paid for much of it anyway as we wanted ABA support).
It might have worked out ok but as they were under no obligation to meet his statement we couldn't take the risk.
What matters most is the suitability of the school and that ds gets the support he needs. No point in being in a fantastic school with small classes if he is sitting in a corner on his own not participating.