Advice from cab for your legal rights is a good idea
The legal position is immensely complicated, as there's a web of inter-related contracts. There's five parties involved - Humblebug, the neighbour, IDNet, PlusNet and BT Openreach (if we bring in the broadband side of things as well, there's also BT Wholesale and possibly another wholesale provider if IDNet don't use BT Wholesale). Obviously, Humblebug is a third party to most of these contracts, so, apart from the limited circumstances covered by the Contracts (Rights of Third Parties) Act 1999, cannot derive any benefit from them. In practice, this means that if IDNet followed the terms and conditions of their contract with Humblebug, plus any terms implied by the law (notably including the term implied by section 13 of the Supply of Goods and Services Act 1982 that services are carried out with "reasonable care and skill" and those imposed by the regulations on transfer of service), Humblebug may not have any remedy in contract.
The remedy to this matter from a legal perspective may well lie within the law of negligence. Once you step from contract into tort law (the area of law that includes negligence), the remedies available are more limited than under contract law and establishing any claim is often be more complicated. Instead of clear rights typically based on a written agreement, as in the majority of contract scenarios, a claim in tort usually involves establishing a duty of care was owed and that this duty was breached.
What remedies are available exist will depend on who did what - which may be hard to establish in this case, as PlusNet may be reluctant to talk to Humblebug and BT Openreach are not consumer facing. It might be uneconomic to consult a solicitor in this case - even a fixed-fee interview for advice will probably cost approaching half the £125 + VAT IDNet want for reconnection.
CAB advisors are not trained lawyers - they're very well trained, but there's only so much they can do. I had a quick glance at the CAB's web site
- their advice on slamming
that has taken place is simply to complain to both service providers involved and possibly Ofcom, whilst any mention of tort (personal injury, negligence etc.) I found led to the advice to contact a solicitor.
A better option may be for Humblebug to take advantage of any legal advice service (s)he may have access to as part of an insurance policy, union membership, workplace scheme, services offered by a bank account or similar.
PlusNet are typically quite active on these forums. If this post doesn't attract their attention, maybe a post in the PlusNet forum here will. Ultimately, it would be much easier if some sort of consensual solution were found, rather than resorting to legal action.
As MrSaffron says, it's probably better to arrange for restoration of service, then seek recovery of the costs involved.