Had she already registered for it online? If so, it says there:
A new 12 month line rental contract applies.In which case the BT phone call would be a continuation of that transaction that you aunt started and would not need to repeat the conditions.
I think this analysis is very questionable.
If BT intend to be bound by the terms on their web site on unconditional acceptance by a customer, those terms are an offer capable of creating a contract on unconditional acceptance. Otherwise, the web site terms are a preliminary stage in contract formation called an "invitation to treat".
Adverts, including those on web sites, are often an "invitation to treat". However, these web site terms seem capable of being an offer: they would not create a supply issue (there's an unlimited supply of Privacy with Caller Display packages) and the terms appear to be sufficiently precise. The only element of doubt in my mind that the web page constitutes an offer is the use of the term "pre-registration".
If BT were using web site pre-registrations merely to call users back to enter into a contract, that would make it all but certain the web site is an "invitation to treat": an indication of the basis on which BT wishes to deal. In any invitation to treat scenario, the contract is formed solely on the basis of a binding offer subsequently made by one of the parties and unconditional acceptance of that offer by the other party. The invitation to treat is not incorporated into the contract.
From a contract law perspective, it is therefore incorrect to say the web site terms could be a continuation of a transaction concluded over the phone. Either the contract was concluded on the web, or it was concluded solely on the phone. As the original poster appears to indicate BT cold called the customer, the contract was concluded solely on the phone, seemingly without an invitation to treat based on the web site..
Whatever the contract formation scenario, any term is only incorporated into the contract if the accepting party had sufficient chance to be aware of it before agreement was reached. If a 12 month minimum period was held to be a particularly onerous term, it would only be incorporated in the contract if BT took explicit steps to draw it to the customer's attention before the contract was formed.