Quite right, it's not a job you can walk straight into - if you started training (with some GCR experience already under your belt) for the grade now, you might be passed out by midsummer! However, if you're interested in the longer-term, it'd certainly be achievable in a couple of years with the right approach - the place to start would probably be the platform grades (Porter, initially, if I'm not mistaken) to get a real feel for how the Railway operates. After a year or so, and probably after shadowing Guards and loco crews for a couple of turns, you could then put yourself forward for consideration for the next training course as a Signalman (seven days of theory spread over a fairly lengthy period, interspersed with practical training).
I'd be happy to help as much as I could (as the Signalling Instructor) to help you through the process and answer any questions or direct them to the right person, but you'd really have to take the lead by contacting the Booking Office - 01509 230726 - and expressing an interest in volunteering on the station (Loughborough or Rothley will probably give you the best experience relevant to signalling, though Quorn's a cracking little station too).
Although I believe we're currently short of TTIs, it does seem to lend itself to going on as a trainee Guard - from there it's almost a sideways step to training as a Signalmen, so it's a bit of a convoluted route if that's what you want to achieve.
It's certainly a bit of a slog, but we'll give you all the training you need, so as long as you're prepared to put the time and effort into learning it (and that's why the experience is essential), you shouldn't have a problem!
Good luck...!
Tom