In the 80s and 90s in the UK you just signed. The card imprint was embossed by the machine which put it onto the carbon, and that was sent off to the bank for payment later in the week, like cheques.
I last used a carbon credit card in the early 00s. Electronic swipe and sign for a credit card had gone pretty much everywhere I went except the US by about 2010
My employer in the UK had a machine in around 2014, but it was only used for sales of their own products to employees.
It put all the transaction risk onto the employer, and had a high fee per-use, but since they only had these 'stock clearance' sales to employees once a year it was fine.
I last used a carbon credit card in the early 00s. Electronic swipe and sign for a credit card had gone pretty much everywhere I went except the US by about 2010