I've tried to find out about her too. She's been there a long time but this is all I could find (taken from the Greater London Industrial Archeology Society's website):
"She is on the south bank just downriver of the entrance to the Canal Basin. From the land the vessel is obscured by large sheds used for metal working. She appears to be a small steamship built c1943, hull part riveted and with engines aft, perhaps a tanker to bunker warships in dock. It has been said that this small ship spent her entire working life at Chatham and also that the original reciprocating steam engine has been replaced by a diesel. Again at one time it was hoped to use the ship for commercial trading purposes. She is still in the colours of the RN auxiliary services. Said to be owned by a Frenchman with an oriental wife — at one time they lived on board which was difficult owing to the inclination of the deck. The ship carries no name or number and is hard to identify. There is a vague memory that at one time she did have a number, perhaps preceded by a letter or letters (a 'C' perhaps), but if so this identification is now completely invisible.
It is said the hull plates are in bad condition and now the vessel partially floods at high tide."
That's all I know, she is hard to get close to to have a proper look.