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Welcome to the forum. If the payment was taken by direct debit, your partner can simply tell his bank that it was incorrect and they must refund the money. Once that's done, please return here if it doesn't end the matter.
If you've got a service with Sky, just ignore the BT message. Tell the building society they're wrong and demand a refund - they must refund it to you immediately then chase BT to get their money back. See bullet point 3 of the Direct Debit Guarantee.