Any question answered...

how feasible is it to put as wide/knobbly tyres as clearances allow on a road bike and go round, say, bewl water http://www.bewlwater.co.uk/page.php?section=2&page=4 or other woodlandy/tracky routes?

Will carbon seatposts/forks/steerers/etc immediately fall off?

Track conditions currently are very muddy and uneven...Cyclists should use cycles built for off road cycling when rides are open.

Does this mean that when they are dried out you can use your hybrid/road bike, or that they always recommend MTB? I'd guess the former, based on other family-oriented parks. Either way, you can ride a Roubaix-qualified* road bike pretty much anywhere, although it might be slower than an MTB and you might want to walk some technical sections. I'd avoid using any real weight-weenie stuff (anything which comes with a rider weight limit, light rims with thin tyres etc.), but I ride my 585 on gravel and dirt paths on RS80-C24 wheels with 700x28C Vittoria Rubino Pro Tech, and it would be under the UCI weight limit if I took one of the bottle cages off.

Your limiting problem is likely to be how much tyre you can squeeze into the frame and fork, not many road frames will take >30C, but not many tyres <31C have enough tread to be useful in mud.

*Doesn't need Specialized decals, whatever they say.