Public shock at Gascoigne's condition at the event on 31 January prompted his admission to a clinic in the United States for treatment for alcohol addiction.
Scotland Yard said Wanstead Flats was the "only practicable site" for its plans and promised it would be returned to its original condition once the event was over.
The person who initially found the envelope is being treated at a Phoenix-area hospital for a pre-existing condition and stress from the event, and others in the immediate vicinity were examined as well.