It does, but international law doesn't work like national law. Who are you goig to sue? The British Museum? The United Kingdom? The Crown? Under whose jurisdiction? Sure, you can prove that the British Museum outright admits a given artefact's origin but how do you establish rightful ownership? It's not enough to prove that something was stolen, you also need to prove who the rightful owner would be, possibly generations after the initial theft.
And if the British Museum (or the Crown, or the UK) flat out says no, what are you going to do? The US infamously established that it will invade The Hague if any US citizen is ever put in front of the Human Rights tribunal, but even without such blatant threats it's difficult to coerce anyone under international law if their own nation refuses to comply. You can try to propose sanctions but there's no "international community", you just have a whole bunch of politicians representing different nations voting on things and they likely don't care about those artefacts enough to do anything especially things that would damage their already subdued economies.
No retrospective application. Hague convention 1907 predates ww2 postdated most acrimonious BM acquisition, noting they repatriate some things (human remains, religious and culturally sensitive objects) and not others (Elgin marbles, bought off the Turks, disputed by the greeks) and the Benin bronzes.
