Single skin wall on Victorian mid terrace

You would most likely have to go through planning and have a structural engineer come and give you some calculations for the footing you would put in I’d imagine you would have to put dowel bars into your existing footing and go from there, for the cost of this project it would most likely be more than it’s worth to you as a customer if its lack of insulation your unhappy with I’d maybe render the property which would make more of a barrier to the climate and weather reducing the risk of damp and install some insulated plasterboard or cork plaster without taking up to much space internally. But it’s highly unlikely that the wall is single skin it may just not have a cavity

I’d look at external insulation first, as you won’t lose any room space. Internal insulation is another option, but it needs done properly to avoid damp and condensation. Costs depend on the size and condition of the wall. Check with your local council regarding planning permission before doing any external work.