Hi Marion, I have lived in SE London all of my life. I lived on the adjacent street to St Mary's Street when I was younger. There is a graveyard at St Mary's Church at the end of St Mary's Street, but I doubt that your ancestor would be buried here unless he/she was affluent.
Most people would have been buried in other municipal cemeteries. There are several of these in the vicinity of Woolwich. There is a cemetery at Charlton(about 2 miles away) but I know that this was a private cemetery when your ancestor died. (it has since been taken over by the local council)
A more likely place would be Woolwich and Plumstead municipal cemeteries which are located in the London Borough of Greenwich. This is the web link that you will need to find the addresses of these cemeteries:
http://www.greenwich.gov.uk/cgi-bin/...RYS+cemeteries+
In the 19th century lots of people in this area were not wealthy enough to afford a headstone. Indeed most people were buried in public graves. This happened to my great grandfather in the 1920s. As a journeyman tailor I had expected that he would have had some type of memorial. However, after writing to the Cemeteries Department they told me he was in a public grave in Woolwich Cemetery. At the time of his death he was living in Charlton, not that far away from St Mary's Street so my guess is your relative will be buried here in either Woolwich or Plumstead cemetery.
A further place that could provide details of your ancestor is the Heirtage Centre which is a relatively new facility in Woolwich. Here you can find out info on the area that your ancestor came from. There is a very interesting exhibition which traces the history of the Woolwich area. Also you can research burials by parish.
http://www.greenwich.gov.uk/Greenwic...eritageCentre/
I hope that this info might help you in your quest to find your ancestor's grave.
Andrea