According to the chronicler Orderic Vitalis (1075 – c. 1142), most of the royal treasure was recovered:
The dwellers on the coast, as soon as they ascertained that the
reports of the disaster was well founded, dragged to the shore the
wreck of the ship, with the whole of the royal treasure; and almost
all that was in the vessel, the crew and passengers excepted, was
However, without any details on what the specific items in that treasure, it is impossible to know if any pieces survive to this day. As for the wreck itself, some of the timber may have been reused or simply used for firewood - we simply don't know.
It seems very few bodies were recovered, though, and William's was not among the few that were. Orderic Vitalis again:
Active men were diligently employed on the seventh of the calends of
December [25th November], while the faithful were celebrating the
feast of St. Catherine, virgin and martyr, in searching along the
coast for the bodies of those who were drowned, but finding none,
they lost their expected rewards. Rich lords caused diligent
inquiries to be made in all quarters for good swimmers and experienced
divers, and offered them large sums for any bodies of their deceased
friends which could be recovered, in order that they might be buried
with due honour.
Orderic Vitalis was one of the more reliable chroniclers of the middle ages and there is little reason to doubt his account. Another chronicler, William Malmesbury, gives a similar - if slightly more colourful - account:
The calamity was augmented by the difficulty of finding the bodies,
which could not be discovered by the various persons who sought them
along the shore; but delicate as they were, they became food for the
monsters of the deep.
The search for bodies would have been motivated by the desire to give them a proper burial than to retrieve any valuables they might have had on them.
Even if some the wreck was not dragged ashore, it's unlikely that any of it has survived:
The strong currents and tides in the area may well have washed away
much of the rest [of the wreck]...while shipworms and other aquatic
organisms would have reduced the wooden parts of the ship unless it
was quickly covered in preserving silt.
The above source also states that
there is no record of anyone having mounted a search or exploratory
and nor have I found any evidence for any search or dives.