I'm exploring the possibility of soundproofing in our home. The party wall, though double-bricked, lacks a cavity. Even in moments of silence, neighboring conversations are audible. The TV volume often needs to be raised significantly to drown out their voices. Additionally, loud door slamming, particularly by children, resonates throughout our house. Despite impending neighbors' relocation, my concern is our susceptibility to noise. Considering a future extension, I'm unsure if soundproofing measures can truly alleviate these disturbances or if relocation is the only viable solution [Link: acousticpanels.co.uk]

Acoustic panels, like you linked, are only good for controlling sound inside that room. For reducing echo, etc.
They are not for controlling sound transmission throughout a structure.

If you have drywall, using Quietrock products, or similar, on top of existing drywall is a great way of controlling sound transmission through a wall. Floor is another matter though.

You can also look at stuff from [Link: kineticsnoise.com]

The best approach is to build a second wall that does not touch the original wall. The new wall should incorporate proper soundproofing techniques. There is no shortcut to following proper technique.

sorry, not in the UK so not sure what products are there or code there.

But to add to the two others. If you can afford to lose some space, build a wall in front of the existing one and insulate with sound dampening insulation. and use two sheets of sheetrock. Also don't forget about the ceiling and floor cavity between the two places, sound will pass where it can, so making a great job on the wall and leaving it like an open window in the floor or ceiling won't solve the issue.

Adding on, you will not be able to sound proof. You will be able to reduce sound transmission. Will it be enough? Depends on what you do.

if you expect or promise soundproofing, you will be sorry.  “Sound reduction” may be more accurate.