Note to self
It's amazing what comes to you on dull train journeys.
In case anyone else gets caught on the same bit (or indeed cares in the first place):
Where I was going wrong is that you're supposed to calculate the likelihood of a particular path, not of a particular combination of v's and w's and so on. It just happens to be a function of the v's, w's etc.
So find density for first transition time :
(mu+sigma)exp[-(mu+sigma)t_1],
multiply by probability that transition goes in the right direction (say able to ill) sigma/(mu+sigma),
multiply by next transition time density etc...
BTW, I really don't like the "probability function" they give for the two-state model p10 section 4. It's not a density function and it's not a mass function; the function's meaningless until you specify where it's which. You also need to specify the range of possible values (outside which the density/mass is zero). So I think the "simple form" is deceptive to say the least. Let alone the "joint probability function"...
For example, say I have a mixed distribution on [0,1], with "probability function" f(x)=1/2, 0<=x<=1. What's the distribution?