Am I insured (in the UK) if I drive down a road marked “no through road - local access only” even if I don’t actually need local access?

If you ARE insured then you can drive anywhere.   

The sign is there only to warn you that you cannot get out at the far end of the road, and that it is only useful if you want to visit somewhere along it.  (It is no use, in other words, if you are trying to use it as a short cut to somewhere else).  It is not meant to restrict access if you are stupid enough to drive down it anyway.  It has no effect on your insurance.  

Insurance for what to cover what?.  A ticket from the police?