[Undertoad]

I couldn't find any such recent BBC story on the BBC website.

I did find that according to the BBC,

Only the museum staff knows what was in the galleries when the war started. The claim that 170,000 items were destroyed or looted has long been abandoned, and reduced considerably. Also, many items have been recovered. Museum staff say that only 33 major items, and around 2,100 minor items, are missing, while 15 major items in the galleries were seriously damaged. These include the famous 4,500-year-old-harp from Ur, with its fabulous golden bull's head.
...
The storerooms tell a different story again. For many weeks outside observers were kept from seeing them. Dr Jabir would only say they had been looted. Even Matthew Bogdanos, the New York District Attorney and US Marine colonel based at the museum, and heading an investigation into its looting, had trouble gaining access, as US policy was to co-operate with the Iraqi museum authorities, and not to behave in too heavy-handed a manner. Bogdanos operated with admirable restraint, considering the US Army was being increasingly held responsible for what had happened at the museum, but it was clearly in his interest to establish how much had been destroyed, who had conducted the thefts, and how to track down and recover stolen items.

Having persuaded museum staff to allow me access to the five on-site storerooms, we all had something of a surprise. Three were still locked, and looked untouched. The remaining two had been entered, with one not even having been locked. These storerooms were generally not ransacked, but clearly some items had been stolen. It seems that the thieves knew where the most precious objects were, and had made straight for them.

BBC November 03

Artefacts looted from the Baghdad museum following the fall of Saddam Hussein's regime have been recovered in London, police have confirmed for the first time.

The discovery of the items follows enduring images from the museum of smashed display cases, empty vaults and crying staff, when reporters gained access for the first time.

First estimates of the looting suggesting that more than 150,000 items were missing were wildly inaccurate.

The most recent figures indicate all but 10,000 have been recovered.

The Cellar awaits your cites.