I believe that chess has a place in every school for children from age 7 upwards.
If you are interested in introducing chess into your school, congratulations. But first I'd like you to think about what exactly you want to get out of it.
Primary schools can look at chess in two ways: as a powerful learning tool from which all children can benefit, and as a game which gives children a chance to compete against each other, in tournaments, and in matches against other schools. But bear in mind that only a few children of this age will reach the level at which competition is worthwhile.
I believe that, if schools see chess both as a learning tool and a competitive game, they need to separate the two. If you try to do both at the same time you'll end up doing neither very well.
The former is best done within the curriculum so that all children can benefit from learning the game. The latter is best done as an after-school club. If you do both, only allowing children to join the after-school club once they have completed the chess course on the curriculum you'll have the best of both worlds.
Chess is an amazingly rich game which can be approached at many levels. It is for this reason that I believe that chess, taught in a structured, step-by-step way, deserves a place on the primary school curriculum. Firstly there are enormous extrinsic benefits, not just through developing cognitive skills, but also developing a wide range of life skills. Read the book and find out how much Sam and Alice learn through their chess lessons. Then there is the enormous opportunity that chess offers for cross-curricular links. Think of all the maths you can do with a chess board and pieces. Look at the opportunities the game gives for both creative and formal writing. Chess can link into history through stories of kings and queens, castles, knights and battles. The game offers amazing opportunities to design chess pieces or costumes for living hcess games in art.
What I would suggest is a one-year course on these lines, run in Y3 or Y4, and taught by a teacher with an interest in chess, with a test at the end. Children who pass the test would then be able to join an after-school club where they would receive intermediate level instruction and be able to play competitively, by playing matches against other schools and taking part in the UK Chess Challenge. When children pass through this club successfully, they would be able to take another test which, if they pass, would qualify them to join a local centre of excellence such as Richmond Junior Chess Club, where they would receive higher level tuition, maybe from an International Master or Grandmaster.
If you're interested in developing a course like this in your school I'd love to hear from you. If you're within easy reach of Twickenham I might even be able to help you myself.