Going out on a limb here, but The Snowstorm by Beryl Netherclift?
Farthingales is Aunt Amethyst. The children agreed on that. She was
trying valiantly to save the gabled old house in the English
countryside, which was beset by taxes and falling into ruin. From the
first day of their visit, Caroline, Richard, and Kit were determined
While it doesn't have "Chronicles" in the title, The Circle of Magic series by Debra Doyle and James D. Macdonald fits plotwise. In the first book, School of Wizardry, we encounter Randal, a squire in his Uncle's household, training to eventually become a knight, who instead goes with the wizard Madoc to learn magic at a school which, yes, is in a castle. ...
Under the heading of possible conflation of sources, Scholastic Press put out The Secret Kingdom in 2011, first book of Chronicles of the Red King, which is about the ancestor of Charlie Bone, who was sent to a magic school in his first book, Midnight for Charlie Bone.