Awful extension next door to my house ...

Building control is very good in my area.
I'd also say give them a ring.

Meant to add, ring the Building Control Department of your local council. The BCs have more power once the building has started and should have done regular inspections. It doesn’t sound as if they have.

1. Party Wall Act RequirementsIf you are building directly on the boundary line or excavating foundations within 3 to 6 metres of your neighbour's property (depending on foundation depth), you are legally required to serve a Party Wall Notice.Types of Notice: You need to serve a notice for a Party Structure (if attaching to a shared wall) or a Line of Junction (if building a new wall adjacent to the boundary).Timelines: You must serve notice at least two months before work begins.Consent: Your neighbour has 14 days to respond. If they consent, work can proceed; if they dissent, you will need to appoint a Party Wall Surveyor to draft an award.

This describes the requirements for a Party Wall Agreement, which your neighbours don’t seem to have.
As said before you will need a solicitor, as it will be the only way to sort it.
Also I don’t think they will have a Building Regulations sign off, as the Building inspectors would have spotted the issue. They may not have needed planning permission as it could have been built under Permitted Development rules, but they will have needed a BC sign off.

Any construction needs to be at least 1m away. You are going to need a surveyor and/or solicitor.

Do you mean the side of your house?

If it’s a party wall to a house, then he would have needed a Party Wall Agreement. You imply that it’s to a garden wall not the house which could be quite different. Is the wall located as yours on your deeds? If it is then I think you will have to get a solicitor involved. Do you have legal representation with your house insurance, the service should be free. If so then a quick consultation with them might be the best way forward.

I too cannot picture this, it does sound horrendous, however you firstly need to go back to your deeds and check that, secondly take a look at council planning portal and check what’s he done is what he has permission to do.

As the previous poster said, check light etc.

Good luck

I can't picture this at all. Is his wall higher than yours? Is your garden wall a wall that separates your garden from his, and one end of it now has his extension 'growing out of it'? Or something else altogether?

I think you need to look into things like being overlooked and whether your light is being blocked, then see if there is anything you (or he) can do to mitigate that. Presumably he had planning permission for the extension? If so, did you see them and agree?

If you can give us a bit more information, we should be able to help more.