My blunt opinion:

Your child is only three and secondary education would not be for another 8 years so wouldn't worry about Private Education at this point with regard to moving to Switzerland.

I would send your child to any Swiss State school as the standard of education and environment of education is better than most UK schools including Private Schools.

Your child will intergrate and socialise better in Switzerland if with the locals. If you continue to speak English at home; your child will have the advantage of being able to speak good English and French or whatever region you are in.

Your child will learn English at a Swiss school as part of the curiculum.

Most foreigners (unlike the English) understand grammer which makes learning languages and writing gramatically correct.