This is one of the lines that I know quite well and have worked over in the past & no it wouldn't be hugely expensive either. The last costings were in the region of £30-50m for the electrification of the two missing sections.
However as the line hasn't had any serious infrastructure upgrades in over 40 years, you'll need to factor in the resignalling as well, which will probably mean an overall cost of £50-60m to give the line a complete update and soon NR will have the perfect opportunity to do it, when the line gets transferred to Basingstoke ROC & Three Bridges ROC.
They need to extend the platforms at most stations, increase the line speeds in several key sections (there's absolutely no reason why linespeeds in excess of 75mph shouldn't be possible), shorter signalling sections should be easily achievable using the lightweight stuff that they used in the Fens.
Power supplies in difficult locations away from roads? You're having a laugh aren't you?
Most locations are quite easily accessed from either the local road network, a substation at Shalford Junction or Shalford station easy access, another sub at Albury (Brook)/Shere Heath again easy access, another at Dorking West or Deepdene which could be used to power both NDL and supplement the Horsham line & final one at Buckland (Betchworth) again accessible from either Rectory or Lawrence Lanes. Simple easy access and TP huts or DC containers are now prefabricated and easily transportable to their corresponding locations and assembled on site.
The other direction you can have a feed from the Substation at Aldershot North, you then already have a Sub at Farnborough for the SWML, another Sub at either Sandhurst or Crowthorne, before joining the already electrified Reading - Waterloo section at Wokingham, where there's already a substation.