Running any part of the national network solely using enthusiasts really isn't practical. Drsatan's points are entirely relevant - although the p-way inspection and maintenance duties might not be such a problem as long as there was sufficient volunteer manpower (at least there's some flexibility, timewise, in those duties), competent cover for attending p-way and S&T faults would be much more difficult to provide on a near-24/7 basis. Likewise, rostering at least two competent Signalmen per box per day, with both turns entering into unsociable hours, would prove almost impossible. The same would apply to traincrew.
I speak from my experience on the GCR - where we (Signalmen) don't have particularly early starts, but we do regularly run as late as 2215 (and sometimes later - I think our latest booked turn each year runs until 0045, on New Year's morning). Trust me, those late turns give our roster clerk a hard time, and that's only one or two late turns per week. I suspect even 2215 is early compared to the finish that would be required on many routes that we might be considering here. Some of our (steam) loco crew turns start around 0600 or earlier (not sure, to be honest) - again, that's a relatively late start compared to the national network. The first Lymington train is in passenger service before 0600, isn't it?
P-way and S&T cover is equally a problem. We, in common with many large heritage railways I suspect, employ a small p-way gang - entirely necessarily. The S&T is, to their credit, entirely run by volunteers - but with a very small number of dedicated people competent to attend S&T faults. In our case, that cover certainly wouldn't extend to a full operation outside weekends, but is sufficient for the relatively light traffic during the week.
Mixing paid staff and volunteers often isn't the most successful exercise. You'd really need to be able to cover most of the essential work with your paid staff, otherwise you'd be in a real muddle if there was no volunteer availability on a particular day (not a huge stretch of the imagination when you consider that most will have day jobs to see to!). They can probably work in the office or something when they're not required for traffic, but it's a bit of a waste of competent, and therefore hopefully relatively well paid, staff. There's bound to be resentment if the volunteers work the 'cream' in the busier and more sociable hours, leaving the rest for the paid staff.
And how do you maintain the competency for the number of staff that would be required? We've got 50 Signalmen to cover between around 8 and 12 turns each weekend, plus a further couple of turns each time there's a midweek daytime or evening service running. If we had to run an intensive service for upwards of 18 hours each day, we'd be looking at 6 turns per day minimum (and that's assuming that the volunteers are willing to work 9 hour turns regularly!), or 42 per week. We'd need getting on for 200 Signalmen to cover that reliably. It's bad enough arranging training and assessment for 50 volunteers, never mind more!
I think there's still a strong need for volunteers on 'community' railways - I'm sure it'd be easier to cover booking office opening hours, for example, or just generally look after the stations. In either case, the service can still run if a volunteer can't be found one day - and it might also avoid some of the worst of the unsociable hours. I don't think it'd ever be possible to operate a full public service from the early hours until late in the evening in that way though.