Hi Dilly
Very difficult to say without knowing the layout and the construction, but, for airborne sound, a great deal can be done by simply adding to the existing walls. I guess the walls in question are just the ones shared with your neighbour?
If so, extra soundcheck plasterboard can be fixed to the wall and finished in the normal way.
Also, acoustic mat, sound reducing felt roll, and other similar products can be used.
A whole new timber framed wall could be added to the existing, and stuffed with acoustic fibreglass.
The plasterboard finishing could be a dry-jointed system, then decorated. This would leave out the wet plastering which can be messy.
With some decent tradesmen, this could be done with a minimum of disturbance. The room needn't be cleared - just kind of moved to the other side - but you would need some guys who were willing to co-operate.