I cannot understand why you are suggesting Consumer Direct, because this organisation no longer existed after 30th March 2012, the government have transferred the work over to the CAB?. If you use the old Consumer Direct website link you will be redirected to http://www.adviceguide.org.uk
I would perhaps suggest that your contract is with the seller of the furniture, and as such it would be their responsibility to deal with the problem. The seller as the contract with the transport company, not you, unless this was done privately between you and the transport company.
I would also suggest that you check the seller's and transport company's terms and conditions if possible. This might give you better leverage for a claim?.
If you paid by credit card, then contact the card provider, and inform them about the situation. they should be able to offer further advice.