You may find some useful information on Broadband Consumer Rights at https://www.broadbandchoices.co.uk/guides/help/broadband-consumer-rights
Above all, and I do know how hard this can be, keep your toys firmly in the pram. The urge to rant and rave is overwhelming when dealing with this kind of problem, but ultimately you need to keep everything on an even businesslike level. If you need to scream and shout take a bottle of cider to the park and yell at the pigeons - you'll make lots of new friends and no-one will care.
Keep records of everything - phone calls, online chat logs especially, emails..... One interesting point in the consumer rights stuff relates to speed. If they've said your line will give between 4 and 8 Mbps (for example) and they are delivering nothing then that is perfectly legitimate basis for cancellation based on the above.
Do seek outside help from the sources mentioned.
If you do jump ship with a faulty line then there's no guarantee that a different ISP will do any better or indeed if they will even take you on. If, for example, your problem is down to connection to an old aluminium circuit then no matter who you use the same issue will apply.
Right now - as a paying customer who is being denied the service for which they are paying, you are in a better position to get this fixed by OpenReach through PN. It is not up to you to provide the solution, it is up to PN to deliver the service they have contracted to provide. Just keep asking them to honour their contract.
I had a similar line issue a while back. Engineer would go to Exchange, reset something and it would go again. Turned out they were just pushing a lose fuse back into its socket. Some bright lad then decided it would be a good idea to replace the holder - sorted.
I wish you luck.
