Lewisham's schools are the least of a parent's worries here. Nearly every school with a grade is good or better. Only one falls short. None is rated inadequate. The worry worth having is getting a place, since the quality is mostly settled.
Across the London Borough of Lewisham, Ofsted has graded 35 schools and 34 of them come out good or outstanding, which is all but one. Four hold an outstanding grade, one requires improvement and none is inadequate. Another 41 settings in the borough have no published overall grade yet, which tells you about the inspection timetable and nothing about the classrooms. That leaves a parent the problem of choosing between good schools, and that is the nicest problem education in London has to offer.
The council decides who goes where under its published admissions rules, and a house round the corner from a school does not secure a place there, however many times the walk gets mentioned on a viewing. The Department for Transport's modelled journey times from 2019 put the nearest primary about a six-minute walk away on average and the nearest secondary about twelve. Distance still counts for something in most allocations, so look up how far from the gate the last offer went in recent years on Lewisham council's admissions pages, and set that against the actual house you are considering. Parents who find out in April that their twelve-minute walk was thirteen minutes too far tend to remember the month.
Prendergast Vale School is the name worth knowing first, because it turns up on both the primary and the secondary lists and can take a child the whole way through without a second scramble at eleven. St Mary's Lewisham Church of England Primary School carries the town's name, and the two Brindishe schools, Brindishe Green and Brindishe Manor, cover the primary years nearby alongside Morden Mount Primary School. For the years after sixteen, Christ The King Sixth Form College is on the local list. The rest of the borough's schools are spread close enough that most families end up choosing between several good options. A shortage of decent schools is the complaint Lewisham parents rarely get to make.
Schools are one of Lewisham's real strengths, and the hard part of the whole business is the admissions form.