"I assume on the line- though that would be impossible to really know for certain."
Have a look where the centre line of the two houses is (I'm assuming here that it's either terraced or a semi).
The wall that you share with your neighbour's house is a party wall. If they have built the extension onto that shared wall then it is on the boundary line.
If it isn't on the boundary line then you would typically still have a fence between their extension and your property.
"If there wall was the party wall then I assume we would not need foundations for that wall?"
I'm definitely not a builder so, sorry, I can't help there.
"We would just use that wall and insulate on side?"
Again, speak to a builder about how they would go about doing that, but it does seem reasonable that this would be all that would be needed.
However, in this situation you will need to go through the whole Party Wall Act thing. This is an added expense that you may/will need to budget for.
First of all you need to provide a written notice to your neighbour. This is the Explanatory Booklet for the Party Wall Act which explains how everything works really well and is definitely worth reading. It also gives some good examples of how to write a notice:
https://www.lincoln.gov.uk/downloads/file/419/the-party-wall-act-1996
or here:
https://www.gov.uk/government/publications/preventing-and-resolving-disputes-in-relation-to-party-walls/the-party-wall-etc-act-1996-explanatory-booklet
They may agree or they may not. If they don't agree then you need to appoint a Party Wall Surveyor. This can cost you around £1,000-£1,500 (and you pay for this). Prices will vary depending on where you live (London is a lot more expensive than elsewhere).
Given that the work will involve removing their gutters then I would guess that they would not agree to this so you will likely have to budget for the cost of a Party Wall Surveyor.
On the other hand, the surveyor will write up what is called the "Party Wall Award" which, as I said, is a legal document that states what work is to be done and how it is to be done.
If your builders need access to your neighbours property then this will be included and your neighbour cannot stop them during usual working hours**.
Likewise, if the Award says that their guttering is to be removed then, again, your neighbour cannot stop this happening.
The whole Party Wall Act thing can get expensive but it does enable reasonable work to go ahead and protects both sides.
** Interestingly, as long as 14 days notice is given, if the adjoining neighbour is not in the property to give you access then the Act says that you can break in:
8 Rights of Entry
(1) ...
(2) If the premises are closed, the building owner, his agents and workmen may, if accompanied by a constable or other police officer, break open any fences or doors in order to enter the premises.