"Babushka" has one single meaning in English. It is this: www.youtube.com/watch?v=ot3cVY1JESQ I suppose the vague connection to headscarves is, "in the manner of a 'babushka'."

Growing up in Canada, I associated the word "babushka" with the simple headscarf folded into a triangle worn by many rural women. My Irish Catholic mother often wore a babushka to cover her head in Church on Sundays in the 1960s and she called it a "babushka". The first printed use of the word "babushka" was in a Canadian women's fashion magazine in 1938. "The babuska is a sort of peasant hood you wear over your pretty curls." --Chatelaine magazine-- The word referred not just to "grandmothers" but to the women who were part of the immigrant wave of agricultural and industrial workers who arrived in North America from Poland, the Ukraine, Russia, etc. after the revolution. The word 'babushka' is one of the few slavic words (a diffcult language for Americans) that made it into the English language. It is an easy word to remember. Russian nesting dolls are also sometimes called babushka dolls (матрешка).

It is the Russian word for a headscarf.