Has anyone checked the local burial records? Even a "Paupers Burial" would be recorded.
As for throwing him in the river - so a few hours later he floats up by the Houses of Parliament - I think not.
I did have a look in NBIv2 but he's not there. Not surprising as few FHSs bothered after the civil registration came in; and hence the death of William is noted:
Deaths Jun 1841
CORK William Greenwich 5 171
CORK William Dartford 5 66
Which one is him I don't know for sure but the Greenwich district covers Woolwich.
Bryan