How to Reduce Neighbour Noise in a BISF Property?

I am renovating a BISF house in South Wales but have an issue with noisy neighbours. I think this is mostly down to them being noisy but partly due to the poor construction between the properties.

As my photographs show, I have stripped the walls upstairs and downstairs back to the breeze blocks. My first task is to ascertain the construction of the wall – is it breeze block / air gap / breeze block?

Secondly, I need to clarify if;

1) Cavity wall insulation can be inserted in the air gap or will it cause problems with this type of construction?

2) If it can be inserted, will it help to reduce the noise when combined with further insulation?

Party wall in bedroom

My next task will be to apply thick insulation to our side the breezeblock wall, covered with thick plasterboard. Hopefully, a combination of these treatments will allow us to enjoy the house without hearing the neighbours. I am sure other owners have encountered this problem and would be interested to hear how they rectified it.

Also, are there any builders out there who can advise me or other owners in the same situation?

Thank you