Each grave is marked by scroll shaped headstones, chosen by the staff and patients at the hospital.
"In tears this morning at this desecration," Ms Palmer wrote as she posted photographs of the desecration on social media yesterday.
The Met Police has asked for witnesses or anyone with information to come forward.
It is the second such incident this year after a memorial in the cemetery was sprayed with paint and the Australian flagpole cut into on the eve of the Gallipoli centenary, the Mirror reported.
Ms Palmer said she has been received offers to help clean the headstones, and though she has been "touched by your kindness" the Commonwealth War Graves Commission (CWGC) is taking up the task.