Yes, there could be an obligation without needing retroactivity
If Elgin never owned them then neither does the British Museum.
It has been a feature of common law that only a person who has good title to property can pass on that title (unless the goods are a negotiable instrument, such as a banknote cheque or bearer bond acquired in good faith). I don't have any idea what the law on this was in the Ottoman Empire.
Let's just focus on the transfer of ownership to Elgin. If that transfer was invalid then Elgin never owned them and the British Museum could never legally acquire them.
The first legal question is: who owned the marbles before Elgin took them? At the time, the Parthenon was being used as an Ottoman fortress; there is, therefore, no doubt that the Ottoman Empire was in possession of the marbles. That doesn't necessarily mean that the owned them; if they didn't own them they couldn't legally transfer them. Even if they did legally own them, did they have a beneficial interest them or were they acting as trustees of some sort of trust (if there is such a thing in Ottoman law)? If the latter, if the transfer was not in the trustees' interest and Elgin knew this, then the transfer is not valid.
Moving on. There is considerable doubt that the removal of the marbles by Elgin was actually authorized by Ottoman authorities.
First, the only evidence that was ever formally recorded (in 1816) is an alleged English translation or an alleged Italian translation (not in evidence) of an alleged Ottoman firman (also not in evidence). This is called hearsay and all it proves is there is an English document and that Elgin says it is what it is - Elgin is no longer around to be cross-examined so this carries as much weight as a Helium feather.
Further, that document doesn't actually say "We are giving you the marbles":
The document allowed Elgin and his team to erect scaffolding so as to make drawings and mouldings in chalk or gypsum, as well as to measure the remains of the ruined buildings and excavate the foundations which may have become covered in the [ghiaja (meaning gravel, debris)]; and "...that when they wish to take away [qualche (meaning 'some' or 'a few')] pieces of stone with old inscriptions or figures thereon, that no opposition be made thereto".
There is some questions of translation here because “qualche” can in some circumstances mean “all” but we are looking at a translation of a translation of an original that might not have said that anyway.
So, that would be the simple legal position.
However, there are almost insurmountable difficulties in pursuing a legal remedy including issues of limitation periods (if someone has your stuff, you have a limited period in which to make a claim for its return) and standing (Greece, as one of the successor states to the Ottoman Empire, probably has the right to stand in their shoes but not necessarily). The resolution of this issue lies in the realms of diplomacy and politics; not law.