The figures, relating to people in private households rather than in places such as care homes and hospitals, are the highest since estimates began for Scotland in October 2020.
Meanwhile, in Wales around one in 65 people are estimated to have had Covid-19 in the week to September 3, up from one in 110 the previous week, and the highest level since the week to December 23 2020.
In Northern Ireland, the latest estimate is one in 60, up from one in 65 the previous week.
Across England, around one in 70 people had Covid-19 in the week to September 3, unchanged from the previous two weeks.
One in 70 is the equivalent of about 754,600 people. At the peak of the second wave in early January, around one in 50 people in England were estimated to have coronavirus.
The percentage of people testing positive is thought to have increased in north-east England, remained level in London and south-east England, and decreased in north-west England.
The ONS said the trend for all other regions was uncertain.