As a general rule, if the fire had been in your home and you were under insured as per your example then most Insurers would deduct the percentage you were under insured by from the amount you receive. However under a sub section such as contents of outbuildings they often just pay you the amount you are covered for eg in your case the £2000.
There are some of the better Insurers that if you present it to them in the correct way will allow you to add your personal possession cover (If you have it) on top of the £2000. For instance if you have cover for bikes, they may pay these on top of the £2000, if you have blanket cover of say £2000 for unspecified personal belongings away from the home they sometimes allow you to add this say £2000 on top and if say you had your fishing tackle specified on the policy they may allow you to add it on.
The above normallt takes an experienced brain to convince them, do you arrange your cover through a broker by any chance. If you do they should be able to try this for you (It does not work with all companies).
If your neighboor had home insurance you might be able to sue them and their Home Insurance pay out under the Liability Section if you could prove they were negligent in the way they lit the fire eg the damage to your shed could have reasonably been avoided. If they don't you could always try sueing them for any shortfall between what your Insurer pays and what you receive. There is no gurantee this would work as you may not win and even if you did they may have no money to pay you back so if you were thinking of this take good legal advice.
Incidently your home policy will normally have an amount of cover to rebuild / replace the shed and then a seperate section under the contents section for "Contents of outbuildings"
Which Insurer and / or broker are you with?