The Windmill Girls were women who performed in the nude as 'living statues' (tableaux vivants) at the Windmill Theatre in London. Because they didn't move, their shows were not considered obscene. The performances continued through WWII, even during the Blitz ('We never closed').
St Joseph's College in Mill Hill, north London was used as the setting for Nonnatus House for the first two seasons. After it was sold to a developer in 2013, the storyline was created in in which it suffered structural damage and had to be abandoned. The Manor House at Longcross Film Studios in Surrey was adopted as the new home for Nonnatus House.
Gripe water is a non-prescription product for colic in babies. It formerly contained alcohol, sugar, and sodium bicarbonate.
REME stands for Royal Electrical and Mechanical Engineers. The Corps was established during WWII.
It was far more common to use the greeting 'Happy Christmas' than 'Merry Christmas'. Also the name 'Santa' was very rarely used in England during this era because the preferred term was 'Father Christmas'. This was likely changed to appeal to modern day and particularly American audiences.