It was the Bank Holidays Act 1871 that made Christmas Day in Scotland a Bank Holiday. As a matter of interest it was (and still is as far as I'm aware) a Bank Holiday only in Scotland as it was an established day of rest in the rest of the UK.
It may have been a Bank Holiday since 1871, but it certainly wasn't a full public holiday until very many years later. In fact, I can remember the jute mills in Dundee would still be working as usual on Christmas Day as late as the mid-1950s. Traditionally, New Year's Day was always the Scottish public holiday.