The Office for National Statistics has published the Labour market overview, UK: September 2021 which contains estimates of employment, unemployment, economic inactivity and other employment-related statistics for the UK. Comment on the statistics is available here. The Institute for Employment Studies briefing on the statistics provides a useful summary.
The ONS has also published the Labour market in the regions of the UK: September 2021 which contains regional, local authority and Parliamentary constituency breakdowns of changes in UK employment, unemployment, economic inactivity and other related statistics.