In English, 86 per cent of their 11-year-olds reached the expected level 4, with 82 per cent achieving the benchmark in maths and 91 per cent in science.
Nationally, 77 per cent of children who leave primary school reached the grade in English, 74 per cent in maths and 86 per cent in science.
Anji Phillips, Richmond's director of education and leisure services, said:
"Our challenge is sustaining high standards and making sure all schools achieve."
Tim Charlesworth, Wokingham's executive member for schools and education policy, said: "With such high percentages being achieved and ever higher targets set for the future, we certainly face challenges to improve on these figures."
Bottom of the table were Southwark and Leicester, which are both disadvantaged areas.
In Southwark, south London, 39 per cent of primary children and 53 per cent of secondary pupils claim free school meals, compared to 18 per cent and 17 per cent nationally.
Education in Southwark was disrupted last year when contractors WS Atkins, brought in to turn the failing authority, pulled out. Cambridge Education Associates now manages the service.
In Leicester, 67 per cent of pupils reached level 4 in English and 64 per cent did so in maths. Southwark had 68 per cent of pupils reaching level 4 in English and 64 per cent in maths.
Leicester is now planning to extend a pilot project in 19 schools which focused on underachieving pupils in 19 schools.
Steven Andrews, Leicester city council's director of education, said: "The improvement in pupils reaching level 4 in these schools was 7 per cent in English and 5 per cent in maths compared to their 2003 results. " A spokesman for Southwark council said: "There are no easy fixes for raising attainment. Improvements need to be made on the process of continuous assessment for each pupil.
"It is disappointing there has been a dip in our key stage 2 English and science results, but we are pleased to note the gap between our position and national figures is closing for maths."
The top and bottom of the primary tables have remained largely static.
Updated scores will be published later this year when appeal results are known. Final tables will exclude pupils recently arrived from overseas.