Are there any non-government tests that are good for primary children?

The First Maths Challenge is for ages 7-9. See here. 

The Primary Maths Challenge is for ages 9-11. See here. 

They are not the same sort of test as SATs. These are the first rungs on the ladder that lead towards international competitions like the International Maths Olympiad. Circles do not ‘teach to these tests’ but Circles do explore similar problems so if a child goes to a Circle they will find the Challenges and similar problems easier.