So HMO rules apply. Also, depending on where you live then there might be a Landlord licensing scheme. I was a LL for 2 years and the borough I was in actually did run a scheme as per @LeaveMyDamnJam suggests.
Barking & Dagenham, Brent, Croydon, Ealing, Hammersmith & Fulham, Harrow, Newham, Redbridge, Southwark, Tower Hamlets & Waltham Forest all have landlord licensing schemes. Good news for you if you are in one of those areas, there will be a dedicated team that you can contact directly.
To check the electrics - try posting those pictures on reddit - r/askanelectrician and specify that you are in the UK. You get mainly Americans on there but they would still know if it was safe or not, but a UK electrician would know which of our laws and regs are specifically being broken.
For your MP / council licensing team / Shelter I advise organising your evidence as follows - index all your supporting documents e.g. emails and pictures, and create two main documents -
- A timeline of all key events - start of tenancy, date deposit paid, date issue with wiring first noticed/reported etc etc, with ref number of each item of evidence
- A list of all current safety issues with the property, again with reference numbers of the pictures / evidence of each item
If you manage to get a meeting with your local MP / council / shelter, it will help you enormously if you have all your evidence organised like this as you won't get long to make your case, and if they decide to follow up on your behalf you'll be making their job a lot easier.