UK Power Networks are the ones to deal with here, the fault is with their network, and it's their responsibility to fix it. British Gas won't be the slightest bit interested, but if they were all they could do is lean on UKPN on your behalf. Usually the problem is getting them to accept that there is a fault - inaction after getting clear evidence from a logger they installed is unusual. To me this suggests they know what it is and that it's a network design issue rather than a fault as such (and therefore very expensive to fix). If it was a high resistance neutral joint they would have fixed it by now because that causes a lot of other problems, and would be likely to get progressively worse.
On a technical level it seems very likely that you're on a part of the network with a relatively high impedance, if they could simply have changed a transformer tapping down then they would probably have done so already - most likely they can't because then at other times of day the voltage would be too low, not necessarily for you but for some customers on the same transformer. A large overnight load that is on another phase is the most likely cause as others have said, in conjunction with a fairly high resistance neutral, probably not faulty just long and thin. Unbalanced loads on the local HV network would could also cause this, but this is less likely. Basically it sounds like your area is in need of expensive network reinforcement and they're trying to avoid it.
The upper limit is 253 V, it's set in the The Electricity Safety, Quality and Continuity Regulations 2002 and will not be changing due to Brexit.
https://www.legislation.gov.uk/uksi/2002/2665/regulation/27/made This is the piece of legislation that they must comply with to avoid heavy fines, and it's what you will need to use to beat them up with. Essentially you need to write to them with the evidence that they are in breach of their obligations and then take it to Ofgem if they ignore you. Separately you can also take them to small claims court for damage to appliances, etc. but only once you have tried the other methods. For those not familiar with the process, in the UK there is a specific procedure you can use for claims under £10,000 which is designed to be low cost and complexity and which does not require you to have legal representation. They specifically cannot claim for their legal fees if you loose, your liability is limited to their reasonable expenses for time, etc. Often big companies just no-show as it's not worth their time and you win by default, see here:
https://www.moneysavingexpert.com/reclaim/small-claims-court/I'm guessing you're somewhere relatively rural and a single-phase customer? Perhaps with overhead distribution, and maybe even on a split phase piece of network? How far are you from the transformer? Is it large or small, does it have 2 HV bushings or 3? Is it rusty (usually indicates severely overloaded, the paint burns off at a lower temperature than the transformer actually fails)? What's your earthing system (do they supply an earth terminal or do you have a rod)?
How low does your mains voltage dip at other times? If not too low then as an interim fix you could buy and install a transformer that takes ~10% off your incoming voltage for a couple of hundred quid, it's far from ideal, but if it stops expensive things failing it might be worth it.
The 315 V is the most worrying, high enough to cause damage, and also more suggestive of a bad neutral joint somewhere than just a general high resistance. It is possible that they've fixed a neutral joint quietly, preventing the 315 V but leaving you with the ~270 V. Options to force disconnection at high voltage exist, but are a bit ugly.