C2C tyres

Firstly, get as wide a tyre as your frame will take. This will make much more difference on tracks than the make of tyre. Secondly, it's worth having some puncture resistance, how much depends on how you weight the tyre performance against the chance of punctures. Schwalbe Marathon Plusses are bombproof but heavy, and you will definitely notice the difference between these and your Grand Prix. Personally I prefer to accept the possibility of punctures - it's not much work to change a tube - and use a lighter tyre. If you like your Grand Prix I would probably go for the 4 seasons version. But there are plenty of other alternatives.

I did it last year on 25mm Vittorio Rubino's which do have a bit of puncture protection but no more than your Conti's.  I was tempted to change to sturdier tyres but worked on the basis that we had a support driver, so at worst I could get a lift to a bike shop and change tyres if it all went wrong.

Off road sections were well compacted stone and OK on road tyres.  Biggest danger of punctures was broken glass on one of the bitmac surfaced paths.  I'm not sayign MArothon Plus wouldn't be better, but I wouldn't buy a set just for the C2C.

It's a great ride btw, have fun.

The Marathon Plus has 5mm of PU inside the tread, making them virtually puncture proof but I agree they are very rigid to fit. I always recommend Schwalbe tyres and the standard Marathon is a great tyre - I have them on my Audax bike - for comfort and mileage.
If you are going soft-roading then you will be glad of 28mm or larger tyres. On the road the difference between tyre sizes (say 25mm to 32mm) is pretty minimal.

Schwalbe Marathons combine good puncture resistance with relatively easy fitting; I have found Marathon Pluses very tight to fit. The Marathons also seem to roll better. Conti Touring tyres also good. With luck you will get plenty of help if you puncture. Not sure if your bike will take 28 size, your local bike shop can advise.

Good luck on the ride!