Question about Frederick, Duke of York and Albany, and his children.

I've seen a few threads about the sons of George III having kids etc, what I was wondering is this, in a scenario where say Frederick Duke of York and Albany, had a son with his wife in say 1792, where would that son be raised? At court in St James, in Buckingham Palace, or with his parents/ father wherever Frederick was based? And what sort of education could this son expect to get?
 
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