Should the Metal Be Showing Through the Centre of an NHS Crown? Is It Ok That the Crown Does Not Completely Cover the Tooth?

From your description it appears you have a metal to porcelain crown fabricated for this tooth.  If after adjustment the metal has begun to show appearing like a silver filling or "decay" as you mention, the preparation/reducation of your tooth was inadequate.  1mm of space for the metal and 2mm of space for the porcelain are recommended.  The metal collar on the lingual side is done regularly because they can get the sharp metal edge flush with the tooth edge and its high polish both help avoid decay.  If this is something that you feel that you are unhappy with, mention it is nothing like you expected and that you don't want any metal to show above or inside the tooth.  The tooth will need a deeper preparation to solve the top metal from ever showing after adjustment and the inside collar can be covered at the doctors request to the laboratory.  A non-metal crown may be also fabricated instead.

The metal on the top of the tooth is most likely due to adjustments that the dentist did to make your bite even....so you wouldn't be hitting on your new crown harder than your other teeth.  This is most likely a result of your dentist trying to be conservative and not drilling away tooth structure, but when the lab got it to make your crown, there wasn't enough room for adequate thickness of porcelain.  Thus, after the adjustment, he/she went down to the metal underneath.  This, in itself, should not cause a problem.  If you see the tooth easily it just looks like you have a small filling in it.

Regarding seeing the metal around the gum line, many dentists feel that the gums are much healthier if the crowns do not go under the gums.  Others, mostly because of esthetics, always try to put the edge of the crown under the gums.  That is a discussion your dentist might have had with you, discussing the difference.  Again, if you can't see it at normal conversation distance, then it is probably not really a cosmetic problem.  It is actually better that the crown margin is not under the gums.  If you don't like it, then the dentist would have to remove the crown he just placed, then prepare the tooth more, take a new impression, and make you a new crown.  Again, discuss this with your dentist, deciding what is most important to you.

It sounds like the dentist adjusted your crown and now the metal underneath the porcelain is showing.  There must be adequate tooth preparation to allow enough room for the crown materials which in this case are metal and porcelain. 

It is very difficult to make a porcelain to metal crown esthetic.  The newer all porcelain crowns are very strong and can function well even on a second molar. I make many all porcelain crowns in my office out of emax and I do not have trouble with breakage or esthetics.  You might seek a second opinion from a cosmetic dentist to see what your options are for replacement if you are unhappy and your current dentist is unwilling to work with you.