The government has U-turned on its plan to press ahead with all primary assessments in 2021, binning Year 2 Sats and a portion of the Year 6 tests.
However, the majority of key stage 2 Sats will go ahead next summer, despite repeated calls for the tests to be scrapped.
The Department for Education (DfE) said the KS2 tests will help to "understand pupils' lost time in education and support those that need it most".
Teachers: Forget Sats and focus on Covid recovery
KS2 Sats papers in reading and maths, as well as the Year 1 phonics check, will take place next year, the department said.
However, all KS1 Sats, as well as the English grammar, punctuation and spelling test at KS2, will be cancelled in 2021.
Schools can take a "flexible approach" to administering the KS2 tests and phonics check by extending the original timetable by a week, until 26 May and 25 June respectively, the DfE said.
The news that the majority of KS2 Sats will go ahead next year has prompted outrage from both unions and campaigners who had called for the tests to be binned in 2021.
The NEU teaching union said keeping most of the KS2 tests in place will mean they "continue to be the focus of Year 6 teaching".
And campaign group More Than A Score called the move "morally wrong", arguing that the plan for primary assessment in 2021 "still largely ignores the needs of children".
The department added that primary performance data "will not be published" for the second year in a row, but teacher assessment in English reading, writing and maths at KS1, and other assessments at KS2, will remain.
This detail prompted questions from both the NEU and More Than A Score, who urged the government to clarify what it intends to do with the unpublished data.
Explaining the plan for next year, the DfE said: "For primary assessments, KS1 tests in English reading and mathematics, and the English grammar, punctuation and spelling tests at KS1 and KS2 will be removed for one year, in recognition of the challenges posed by the pandemic.
"Schools can take a flexible approach to the administration of the KS2 tests and phonics screening check, by extending the original timetable by a week, until 26 May and 25 June, respectively.
"Although primary performance data will not be published, teacher assessment in English reading, writing and mathematics at KS1, and all other assessments at KS2, will remain.
"These measures will help to understand pupils' lost time in education and support those that need it most, providing vital information to parents and assisting with pupils' transition to secondary schools."
The DfE has since clarified that KS1 and KS2 teacher assessments in science will also be cancelled in 2021.
The multiplication tables check, which would have become mandatory for Year 4 pupils in 2020 if assessments had not been scrapped due to the pandemic, will be optional next year.
Mary Bousted, joint-general secretary of the NEU teaching union, which has called for the 2021 Sats to be scrapped, said: "Since the beginning of the pandemic, schools have been working to care for children's wellbeing and to engage them in learning. They will ask whether this latest announcement helps or hinders their work.
"At KS1 the cancellation of tests and the focus on teacher assessment are welcome. Removing the pressure of tests will free up time for teaching the broad and rich curriculum which children need now more than ever.
"We believe that tests in KS2 should also have been cancelled. Keeping most KS2 tests in place will mean that in many schools they will continue to be the focus of Year 6 teaching. The government has not recovered from its addiction to testing, and children will pay the price. Months of test preparation are not the way to 'build back better'.
"The government needs urgently to clarify what it means to do with test results. It says that they will not be published, but this is very different from saying that they won't be used. Teachers require an assurance that the results of KS2 tests in 2021, which cannot possibly be a measure of the quality of a school, will not be used as a measure of accountability.
"The suspicion remains that KS2 Sats are being brought back not because of their educational value, but because the government fears that schools will learn to live without them."
Geoff Barton, general secretary of the Association of School and College Leaders, said: "We are disappointed that the government has not cancelled primary school Sats. Even though these won't be used as the basis for performance tables, holding a set of statutory tests still represents an unnecessary extra pressure."
Paul Whiteman, general secretary of school leaders' union NAHT, said: "The reduction of some of the burdens in the system will begin to allow school leaders to focus on the quality of education they are determined to provide for pupils.
"Whilst the government has not gone as far as we would have liked, they have moved significantly towards the profession."
Patrick Roach, general secretary of the NASUWT teachers' union, said: "The decision to require primary assessments, including KS2 Sats, to be undertaken is extremely difficult to justify. These assessments will serve little purpose and have no validity for those children whose education has been seriously impacted by the pandemic."
A spokesperson for More Than A Score said: "The new plan for assessment in primary schools still largely ignores the needs of children. After intense campaigning, we are pleased that Year 2 Sats, alongside the much-maligned Year 6 Spag [spelling, punctuation and grammar] Sats paper, have been dropped this year. However, formal tests will still take place in four out of seven primary school years.
"Making the Covid generation of children sit most of their formal tests this year (under GCSE-style exam conditions for Year 6s) is morally wrong. It risks amplifying the harms to learning and mental health already caused by the virus, particularly for the most deprived in our communities. It will produce near-useless data for the government and will tell parents nothing they don't already know."
They added: "Nick Gibb [schools' minister] has stated many times that Sats are designed to hold schools accountable. They are not fit for purpose to measure the impact of the pandemic on children.
"Schools have already carried out their own precise, teacher-led assessments, providing all the information necessary to see where learning gaps are and how to bridge them. The government has yet to make clear how the data it gathers will be used, even if it is not published in the usual way.
"The key reason the government has decided to tinker round the edges, rather than do the right thing, is because it fears that Sats will lose their power completely if they are dropped for a second consecutive year."
How do Sats work in a normal year?
Sats are national tests sat by children in state schools across the country.
At KS1, children are assessed in English, maths and science. They sit four mandatory papers: two reading papers and two maths papers. There is also an optional spelling, punctuation and grammar (Spag) test.
The results are not reported to the government. Instead, they are used to feed into teacher assessments of pupils' achievements.
It is these teacher assessments in reading and maths, together with further assessments in writing and science, that are reported on a national and local authority level.
At KS2, children are assessed in English and maths. They sit six papers: one grammar and punctuation paper and one spelling paper (together forming the KS2 Spag test), one reading paper, and three maths papers. All tests are marked externally.
Children at KS2 are also assessed by their teachers in writing and science. The Sats results in English and maths, together with these teacher judgements, are reported on a national and local authority level.