• StatsWales, Welsh Government
• Statistics and research, Welsh Government
• DataMapWales, Welsh Government
• data.gov.uk, UK Government
• Research and statistics, UK Government
• api.gov.uk, UK Government  This is a list of the APIs in the UK public sector.
• ONS Census data, Office for National Statistics (ONS)
• ONS Local Statistics, ONS
• Other datasets and publications, ONS
• NOMIS, ONS  NOMIS holds similar data to the ONS site. But is worth mentioning as it has an API
• OS Data Hub, Ordnance Survey   This is not publicly available, but is free for members of the PSGA (which almost certainly includes you!)
• Infobase Cymru, Data Cymru   Mostly data which is available elsewhere but presented for a Welsh local gov audience with bilingual metadata
• Self-Assessment Performance Dataset, Data Cymru  Data collection for Welsh councils to support benchmarking and performance work
• Companies House data and the Companies House API, Companies House
• Public Health Wales Observatory, Public Health Wales
• Open crime and policing data UK, Data.Police.UK
• UK Parliamentary data, UK Parliament

Please comment below if you have more to add.