They cannot put this astride the boundary without your consent. They can build up to the boundary but not over it.
Have you agreed to it?
At any stage in the future, is there any likelihood that you would have a similar extension in the same place?
If so, then what you need is an agreement (Party Wall Award) that you can use their wall as the side of your extension and that you can key into it. It would save you the cost of one wall in your extension. I know that you may have no plans at the moment but at some stage in the future, you may decide to do this or you may sell the property and your buyer may do it. Otherwise, if another extension is done at some stage in the future, it means that you are going to lose a large proportion of the width of the property probably as much as 300 mm or even more.
Once it becomes over the boundary, it then becomes a party wall whereas if it is up to line of junction it’s not.
If it is coming over the boundary and there is no agreement for you to key into the wall and to use that wall as the outside of your wall, then your lender is going to have to give approval to this.
What the neighbours trying to do is grab a few extra inches at your expense what you want to make sure is the you get the benefit of what they are doing by giving consent for the wall to straddle the boundary and for you to be able to use it at some stage in the future. In order to do that, there may be considerations in respect of the guttering if it’s going to overhang (it would need an easement) and with regard to the guttering and future development.
Although this may seem straightforward, it is likely to be far from it and hence that’s why you need a Party Wall surveyor to look at the whole situation. The cost of that comes down to the neighbour.
Can I clarify anything for you?
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To answer one of your questions, it won't effect the deeds to your house.