When does (or did) school resume in the fall in the UK?

There is no standard date, but most schools in England and Wales resumed in the first week of September. In Scotland, they normally start a week or more earlier. The last Monday of August is a public holiday in England and Wales (known as the August Bank Holiday), but not in Scotland, and the Scottish schools are normally back around the English Bank Holiday.

Can we narrow this down please? England or Scotland? Which local authority area.

Roughly, the Scottish schools go back mid August and the English schools go back the first week in September.

Are you speaking British or American?

Schools (ie things you attend before 18) in England restarted in early September (the precise date depends on the school and the authority). Schools in Scotland mostly restarted a week or two earlier.
Most tertiary education establishments (collectively known as 'universities') restart nrext week.
Oxford + Cambridge restart in early October.
There is a complicated difference between "term" and "full term"

Specifically, I meant what in the US would be high school and in England. However, your kind replies more than answered the general idea that I needed. Thank you.