Fine, mostly, with a few you would drive around. Across the wider area about half the graded schools rate good or better. Only a handful, around five, carry the top Outstanding grade. A similar handful sit below good. Winterbourne has its own primary and its own secondary and neither is the problem, but the honest headline is average with good options in it rather than a clean sweep.
In the village itself the obvious names are St Michael's Church of England Primary School, Winterbourne for the little ones and Winterbourne Academy for secondary, with Hambrook Primary School and Frenchay Church of England Primary School close by for families on the fringes. Reach a bit further and Frampton Cotterell Church of England Primary School, Bromley Heath Infant School and Bromley Heath Junior School come into range, and for secondary the net widens to Abbeywood Community School, Downend School and Mangotsfield School. The choice is real, which is more than a lot of villages can say.
Naming a school is not the same as getting into it, and that is the part worth being sober about. A school being close, or being good, does not put your child through its gate, because admission runs on catchment lines and distance measured to the metre, and those move year to year. Buying the house nearest the good primary is a strategy, not a guarantee, and plenty of families discover in the March of application season that the tie-break went to somebody who lives forty metres closer. If a specific school is the whole reason you are moving, check its actual admissions for the actual year before you exchange, not the reputation and not the postcode.
Spread across the full area the grades are a mix rather than a promise. The Outstanding schools are the minority everybody wants and everybody's neighbour also wants, the good ones do the bulk of the work, and the small tail rated below good is the reason the average lands where it does. Nobody is stuck with a bad-only option here, but nobody should assume the nearest door is the best one either. The good schools are good. They are just not evenly spread, and the map does not care which house you fell in love with.
Blunt version: the schools here are solid and plural without being a knockout, which is the average result the numbers give and the average result the place deserves. Buy for the school and buy the catchment, not the brochure, and check the admissions before you sign anything.