I appreciate your point here, but this seems completely flawed. When creating or terminating any contract, at the most fundamental level, a wet signature is required, though these days, that is mostly skipped on so many levels. For BT to just issue a fire and forget notice saying services will be transferred is just madness, in my opinion. What if the person is ill ? Away on holiday ? Or post delayed ? That maybe how the current process works, but in my opinion, that doesn't make it right, or even legally valid.
I appreciate the thread has moved on beyond this point, and note that MrSaffron correctly pointed out the falsehood in this at the time. However, the wider context of this answer is relevant to where we are now.
It has never been a requirement of the law of England and Wales (or, so far as I know, Scotland or Northern Ireland) for contracts to be in paper form and to be signed. If there were such requirements, City traders would be in trouble, as their job revolves around electronic or telephone contracts to buy and sell worth millions or billions of pounds. A contract can be verbal - though it can be hard to prove a verbal contract.
A contract in England and Wales typically requires an offer, unconditional agreement to that offer, consideration (best thought of in this case as a promise to pay in return for a promise to provide a service) and intention to create legal relations. There are certain exceptions relating to consideration (deeds, estoppel and so on), but they are not relevant here.
The mention of postal delay by mixt addresses a particularly knotty point of contract law in England and Wales. Acceptance of an offer by post, which would create a contract, usually applies from the point of posting, even if the letter is never received - though, of course, it may be difficult to prove you posted a letter of acceptance that was never delivered. This rule does not apply to instantaneous electronic communication (the case law is about telex, but applies by extension to fax, e-mail and the like) - in that case, acceptance applies from when the offeree reasonably expects the offeror to notice the acceptance.
The concept of privity of contract is central to the dealings between the original poster and TalkTalk. The original poster seems to be a third party to the contract between TalkTalk and the unnamed woman on the contract, so has no rights whatsoever under that contract. The Contracts (Rights of Third Parties) Act 1999 cannot apply, as the original poster was never supposed to gain any rights under that contract.
As others have suggested, I suspect TalkTalk have mixed up the original poster's phone number for the one on which the woman contracting with TalkTalk wanted service. Once that mistake has been pointed out to TalkTalk, it's unclear what will happen. TalkTalk may try to argue that the original poster has entered into a contract with them for provision of service, based on his dealings with them in relation to the matter. Alternatively, TalkTalk may argue that there is no contract in relation to the original poster's line, and may cease all services without notice.
Arguably the original poster made two mistakes.
Firstly, he should have responded to the notification from the losing service provider by stopping the transfer. It's never a good idea to allow a mistaken transfer to proceed - if you wish to switch to the gaining provider, stop the transfer process and start it again on a proper footing to prevent this sort of mess.
Secondly, he should not have attempted to deal with TalkTalk in relation to the mistakenly provided services, other than working with TalkTalk to transfer service back to a provider of his choosing. Making use of the services knowing that they are not provided for his benefit could constitute fraud by false representation contrary to section 2 of the Fraud Act 2006 - though it is debatable whether the original poster's actions are dishonest in the legal sense (dishonest according the standards of ordinary and reasonable people, and the subject realised their actions were dishonest according to those standards). No offence has been committed if the actions were not legally dishonest.
If the losing provider issued the notifications required by regulation at the appropriate time, they are not negligent, and the contract with the losing provider was correctly brought to an end. Contract law does not sit in isolation - if the law lays down a process by which a service is transferred to another provider, and the transfer process was correctly followed, the contract with the losing provider ends on transfer despite the customer's lack of explicit instructions to the losing provider to terminate the contract.
If the losing provider correctly followed their part of the transfer process, it is hard to see how they breached any duty of care they owed the original poster, which is the fundamental requirement to establish negligence. Contributory negligence would be a further hurdle, as it can be argued the original poster was negligent in not stopping the transfer from taking place.
Damages in tort (including negligence) are usually limited to direct financial loss, so there appears to be no route to claim for consequential matters such as alternative broadband provision. Even if consequential losses were claimable for, the concept of remoteness would limit damages to those directly flowing from the loss - the original poster certainly couldn't claim for the whole of a 24 month mobile broadband contract!
I think the original poster has three options to resolve this matter.
1. Instruct TalkTalk to cease the mistakenly provided services at his address.
2. Instruct TalkTalk to transfer the line to a provider of his choosing, on the basis it was mistakenly transferred to TalkTalk.
3. Instruct TalkTalk to set up an account in his name, and transfer service to that account.
I strongly suggest that "do nothing" is the wrong approach, as it may only make the situation harder to resolve in the future.
I agree with the suggestion that any dealings are best done by letter or fax, so as to be traceable. If using fax (yes, I know it's old school), it's important to print out a transmission report confirming receipt.
Nothing in this post constitutes formal legal advice. I am, in any case, a mere law undergraduate, who could well be talking rubbish.