For reference, but the London-wide service from the City of London is for hazardous waste, chemicals etc., not clinical waste.
For household clinical waste in Barking and Dagenham, Havering, and Redbridge you should call Medisort on 0330 122 4220 to arrange collection.
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www.barkingdagenhamccg.nhs.uk
In my London borough the council need an N.H.S. referral to be able to arrange collection. No wonder the surgery just lets you drop them off there, less work for them.
It also seems rather risky as surely people who do their own glucose monitoring rather than through a G.P. (let alone any other reasons why someone may be injecting themselves) will end up having to put their waste in normal bins.
No idea if Medisort require any similar sort of approval or will collect from any household in those boroughs.