I was at school in Blackpool in the 60s and had a friend who used to spend her weekends taking part in ballroom dancing competitions, so I was perhaps more interested in the programme then than most teenagers.
At that time Come Dancing was a competition between different parts of the UK and was broadcast from ballrooms all over the country. I think it was teams from two regions each week, each one providing couples to perform the various dances and a formation team: my favourite bit. IIRC, when in Blackpool it wasn’t from the Tower Ballroom or even the Winter Gardens, but the Mecca ballroom on Central Drive. I remember going to ordinary dances there and drinking Babycham. As I was still at school and it was before exams I must have been underage.
According to this Wikipedia article Come Dancing began by broadcasting from various ballrooms owned by Mecca, which fits with that.