BA will simply ADM BA Holidays for the balance and the hotel may do the same. What BA Holidays does vis-a-vis OP is between the two of them.
Best practice is to call BA Holidays and then confirm with an email that you originally cancelled on a specific date in the past and then called again today. It's likely sufficient if you have a copy of your email from BA Holidays confirming the cancellation. Don't lose that email.
So, bottom line is that it isn't worth the time, effort, and money to deal with this downstream when a quick call and email today will solve it.
As to the taxes, they are refundable. But, to BA Holidays.