It looks like to some extent this is a bad idea. Running with thin disks makes you susceptible to nerve pinches, numbness, and possibly worse. At the lower end of "worse", I suppose you have the extreme pain of the full-blown herniated disk (or "slipped disk"). At the upper end, ... well I don't want to think about it, actually.
A nice refinement to c25k might be to compile a list of "common obstacles" (with a better name than that) for the older runner, restarting after many years of neglect. It neededn't be a comprehensive list (just say "this is not a comprehensive list"), but one of the first things to put on there is that there's probably a different adaptation process someone with thin disks needs to follow, getting into running. There would be various risk factors like this that c25k could be used to learn about. Ah! There's a better name for the list of "bad things" : "Risk factors".
Enough of the generalities. In my own case of mild numb toes the latest progress is that I'm not allowed to exercise, but I'm allowed to walk, and on Monday I go for MRI to help unravel the neurology of the variably numb toes, the brand new sensitive tendons, the even more variably numb little fingers and hands, and the newly discovered numb spot on my back. I could complain, but what would have happened if I hadn't tried running? All this stuff would have built up to be worse than however bad it's going to be now. c25k is a very mild bump to your system, rather than a shock, so anything it rattles loose is something that was going to need attention at some less robust age.
So there will be no weight training this afternoon, there will be no spinning tomorrow, and on Monday, there will hopefully be news that is not *too* bad (being realistic, here). I will do some long walks, though, because the list of things that mess this situation up (from worst to at least the least uncomfortable) is: This desk chair (terrible), the chair in the lounge (not all that bad), standing still, walking, running. If I run I'll feel fine for a good 3 hours or so. While I walk everything stays stable or gets better. When I stand still things get worse (and actually this is worse than the armchair, because I can "dangle" in that - unlike an orangutang, but also not strictly hominid, either). Oh and the bed doesn't help. Run to the hills, run to the valleys ... (Peter Tosh said something like that in Downpressor Man once, I seem to recall)