Sand textured paint on lining paper

there is no easy fix on this, you must sand walls then strip lining off back to plaster, possibly pva and skim to achieve the surface to paint on..

remove with the lining paper treat the walls, smoothen it than paint it with any type of finish emulsion paint you'd like

The only way i would be to remove the wall paper and the sand textured paint at the same time, the paint shouldn't of gone through the paper so there is no reason to strip paint under the paper. Sand walls and start fresh Painting. If its bare plaster under paper still sand walls and then use johnstone's coverplus vinyl Matt as it can be used on bare plaster with out the need to water it down and you wont have to use PVA either. Then apply the paint you want to use. Rob from Bevan,s Painting.

Strip the wallpaper skim and paint. 1 mist coat 1 or 2 top coats depending on the paint quality to achieve the best result.

Only solution is to strip lining paper,only then can you assess the condition of your walls ,.if plaster has perished beyond repair then you would think about re plastering, if walls have minor cracks ,you can fill and sand to good finish ,You must remove all traces of paste from wall,as this will stop you applying filler correctly and also any paste left on walls will make the new paint you apply peel.