Hi OP. As explained, grammar schools have been largely abolished these days. There are some local authorities that have kept the 11 plus system - ie. parts of Kent, Essex, Buckinghamshire, Lincolnshire and Wirral (?). In addition there are some other grammars dotted around - eg. in Devon and some parts of London. I think there are about 220 grammars across the entire U.K. (so not many in the scheme of things).
Don't be confused by schools that have become private, but just retained the name "grammar" - eg. Kingston Grammar. Also some comprehensives may have done the same.
Basically, "grammar" just means children have to pass an 11 plus exam to get in. But not all grammars are equally selective - eg, the ones in Lincolnshire may take up to 50% of local children, whereas somewhere like Tiffin in Kingston will only take about 5%. How selective a school is will be reflected in exam results.
If you are in Brixton, the nearest grammars would be those in the Borough of Sutton (Wilson's and another one, I think)? But I think you need to live in Sutton Borough to apply. Apart from this there is Tiffin Girls and Tiffin Boys in Kingston - these are so-called 'superselectives' with no catchment area. In North London there are a couple of other very selective grammars - The Latymer School ( not to be confused with Latymer Upper in Hammersmith which used to be a boys grammar, but is now a superselective co-ed independent day school) and Queen Elizabeth School. There may well be a couple more.
Grammars are not the only form of selective state schools. Have a Google about Brampton Manor in East London - a very selective state academy that gets more than any other school (including Eton etc) into Oxbridge every year. I think there is another similar school in that area as well.
Other schools are faith schools and have their own form of selection which may well involve some nonsense about which children were baptised as quickly as possible after birth; or who has shown up to help with church raffles etc.
Don't just 'chuck' your kids in any old school. London is not like other parts of the U.K. and people outside London don't always fully understand this. Some schools are rife with stabbings and all kinds of antisocial behaviour and this is why so many families make the decision to go private / move out / move to certain catchment areas. In London, many of the best schools in the U.K. are in proximity to some of the worst. The gulf is huge and reinforced by the fact that those who can will inevitably seek better / safer options for their kids. It's all very well people in the suburbs and country pontificating - how many children have been stabbed at their local schools?
Anyway, good luck.