Marchwood does the early years in the village and outsources the teenagers. There is a Church of England infant school and a junior school within the place itself, so the small ones can be walked to the gate, and the secondary options sit a short drive out toward Totton. It is the classic commuter-village arrangement: primary on the doorstep, secondary somewhere with a bigger car park. Nobody is bussing a five-year-old across the county here. For a village this size, having its own infant and junior schools at all is more than plenty of places manage.
What none of that guarantees is a place. A school being near your front door and a school having room for your child are two different things, and the second is settled by an admissions formula that does not care how short your walk is. Buy on the assumption that your nearest school is your school and you may find out in March that the map and the offer letter disagree. Check the current catchment for the exact address before you fall for it.
In the village, the two names to know are Marchwood Church of England Infant School and Marchwood Junior School, which between them carry local children from the first day of reception to the edge of secondary age. After that the trail leads out of Marchwood, with Hounsdown School and Testwood School among the secondary options in the wider Totton direction. There is also additional and alternative provision across the broader area, including Forest Park School and the New Forest Alternative Provision, for children who need something other than the mainstream route, so the map is not as thin as a village this size might suggest.
A word on the ones you might half-remember: schools close, merge and rename, and a name that was gospel a few years ago can point at a locked gate today. Go by the current admissions list rather than what a neighbour told you or what was true when they were at school.
On the schools, then, Marchwood is solidly middle of the road: enough of them, close enough, without a single standout name that makes an estate agent reach for a bigger number. It covers the basics and leaves the showing off to somebody else.