Can anyone tell me if a UK company who set a payment plan for two years and these payments are made in full in the time agreed,can at the end of this time,with all payments completed say they have made a mistake and claim a further amount to be paid.

As far as I´m aware, you will have signed the payment plan contract which in real terms is legally binding. If you´ve paid in accordance to the contract then there should be no further payments or fees. Citizens advice will clarify.
Mistakes can be costly ....

Look up The Consumer Group on the internet. I think it's a forum like this one but works a little differently.
They are a really useful source of information and your story will have them raring to go and hopping mad if it's happened before they will know about it. If it hasn't they will want to!

I would agree, if you have followed exactly the terms of the plan/contract and it doesn't mention possibility of further payments, unless you have contravened the agreement, then I don't see any obvious reason for them.

Surely they gave a reason of some sort in their request for further payments ???

The reason given was that they had made a mistake in their calculations and given me a repayment plan for three years not two,although the direct debit mandate clearly states final payment 1st December2013
I made it clear at the onset that I only wanted to repay in 2 years.

Looks very much like they are trying to rip you off from what you say.

If your total payments amount to more than the original retail cash price of the article, then it appears you have paid the full amount.

If they amount to less then obviously the company have made a mistake. However from what you say I would say it is still their own responsibility, not yours and they should honour the agreement.

As mentioned, check it out with consumer advice just to be sure and don't let them pressure you into coughing up if it is their own responsibility.

Thanks for your help and advice,I will certainly not be pressurised into making any further payments as a result of their mistakes.

Although, as Dick says, if the sum of the payments you have already paid does not exceed the original agreed price for the goods (plus interest) then you have not completed payment. As your arrangement sounds like a type of hire purchase agreement then you do not actually own the goods until they are fully paid for. If the hire purchase company made a genuine mistake then they are entitled to the remainder of the money - or they can seize the goods back (which legally are still theirs)

The arrangement was to pay for a funeral plan

Don't agree. If both parties have signed up or agreed to a fixed contract and the buyer has completed his/her obligations as per the agreement, I would say that the vendor/HP company is responsible for any error.

Just my opinion of course, so it would be interesting to hear of a ruling in such a case.

You cannot hold someone to an unfair contract - contracts are not absolutely binding. If the lender can show that it was a genuine error - so if for example the goods have not been completely paid for (plus interest and charges) - then it would be easy for them so get a court to agree that the outstanding monies should be paid.
Indeed if it is the case that the full amount was not covered by the term(2 years) - and we don't know that as the OP hasn't said - then it could be argued that the OP should have realised this and brought to the attention of the lender.
if, however, the goods (plus charges and interest) have been fully paid for and the lender is trying it on then the OP is quite within his rights to tell them to go whistle (politely, of course, and in writing)