[I am doing JOGLE in August on a GT Grade adventure bike...]
I suppose your choice of route will have a bearing on your choice of tyres.
My son and I finished LEJOG on Sunday riding Airminal Joeys shod with Schwalbe Marathons (24 x1.75) running at 60psi. We were self-supporting camping/hosteling so had a fair bit of gear. We covered 1220 miles and had two punctures (one pinch, one penetration) - both in rear wheels. The tyres were 'worn in' when we set off - but still had plenty of life left. Had the tyres been new (i.e. 'Plus'), we could have probably avoided punctures all together.
We did a mixture of Sustrans and Cicerone. Although we really enjoyed the traffic-free sections, I have to say that the surfaces of some of the sections were at the limits of what I'd consider suitable for a 'touring style' tyres. And yes, lots of cattle grids. We never have any problem with them - although obviously treat them with caution, particularly in the wet. We took spare inners, but I wouldn't bother taking a spare tyre - particularly on standard size wheels - there are some great cycle shops en route.