The schools in Sway are, for the most part, somebody else's. The village runs no graded school of its own, so families here lean on the primaries and secondaries in the surrounding villages and towns. The nearest primary is a short hop. The nearest secondary is a proper journey. This is a place where the school run is a genuine run.
For the primary years, the options sit in the settlements ringing Sway rather than in Sway itself. Tiptoe Primary School, Pennington Church of England Primary School, William Gilpin Church of England Primary School and the Hordle Church of England primary are among the names within reach, and the closest of them works out at roughly a quarter of an hour on foot or under ten minutes by car. Secondary-age children head largely to Priestlands School, which is nearer twenty minutes in the car and about half an hour by bus. For those going independent, choices such as Walhampton School and Southlands School sit within the wider area. None of that spread is unusual for a forest village, but it is a spread all the same.
A word of caution on all of it. Being near a school and being entitled to a place at it are two different things. Admission runs on catchment lines and criteria that move year to year, oversubscription can shut the door on the school at the end of your road, and a name appearing on a local list is not a promise your child walks through the gate. Anyone buying here with one particular school in mind should confirm the current catchment and the exact year of entry before the offer goes in, not after the boxes are unpacked.
The blunt verdict is that schooling here is fine rather than a feature. There is nothing graded on the doorstep to build a move around, but the towns and villages nearby cover the years perfectly well, so long as you are content to drive them there and back.