Have been looking at Historical Directories on line.
According to the Lincoln Directory, 1913 there are 3 St Botolph's Church schools in Lincoln (Boys - T W Gunton, Girls - Miss E A Pepperdine and Infants - Miss K E Whattam). The Girls and Infants are in Bargate, between Sidney Street and Craven Street.
Can't find a St Botolph's in Boston. According to Kelly's Directory 1919 the school in Pump Square is the National School
regards