According to Wiki, it was double track from Norton Fitzwarren to Bishops Lydeard and from Dunster to Minehead only. The middle section was only ever single track with passing loops. Available photos on google and wiki corroborate this with some stations only being wide enough for 1 track and every indication suggesting it has always been son.
We're quite likely talking 10s of millions to upgrade the line, for reference.
According to Wiki, it was double track from Norton Fitzwarren to Bishops Lydeard and from Dunster to Minehead only. The middle section was only ever single track with passing loops. Available photos on google and wiki corroborate this with some stations only being wide enough for 1 track and every indication suggesting it has always been son.
Quite possibly hundreds of millions. NR or GBR would start by expecting the line to be upgraded almost as if it were a new line. In places it has a winding course with many embankments and cuttings, and present-day standards would probably require gentler slopes. There are numerous level crossings which would require automation of one kind or another.
I don't know what the service was in 1971 when the line closed, but during September 1962 - June 1963 it was nine trains a day on weekdays and three on Sundays. Not enough to cover costs. There was probably a declining amount of freight as well. One train ran fast from Taunton to Williton and reached Minehead in 49 minutes. Others took from 50 minutes to over an hour. So very little faster than a car journey now.
When the West Somerset Railway was planning a heritage line, it thought about a diesel commuter service to Taunton, but that didn't materialise, presumably because it wouldn't pay its way. At one time the WSR tried to encourage visitors to Butlins to use the line, but getting a bus from Taunton to Bishops Lydeard, changing to a train, and then a bus or taxi from Minehead Station to Butlins, would make a through bus more attractive.
Minehead is now a bigger town than it was fifty years ago, and it does serve a large area, though that is sparsely populated. Many of the inhabitants of that area will own a car. If they are going to Taunton, either for shopping there or to get a train somewhere else, it may well be quicker to drive straight to Taunton than to drive to a WSR station and get a train.
As other posters have said, the easiest way to meet the peak demand from Butlins visitors would be a coach from Taunton direct to the holiday camp.
I don't see the Minehead branch re-opening as part of the national rail system until we return to a time when most can no longer afford a car and have to rely on public transport - probably for a much-reduced amount of travelling than most of us are accustomed to today.