The idea came out of the Model Health Inquiry, set up by the BFC in March last year to investigate health problems suffered by models in the British fashion industry, following a furore over "size zero" super-skinny models.
New York, Paris and Milan -- other centres of the fashion industry -- have said the plan is unworkable.
Riva blamed impracticalities in implementing the annual checks, a feeling among models that it discriminated against them and a lack of support from industry bodies in other fashion capitals.
"This will only work if it's an international solution," she wrote.
"If we could just tick a box and move on it would make everyone's life easier but now this really is a long-term behavioural and educational campaign."
The comments were confirmed to AFP by the BFC.
The BFC had previously warned that models based abroad could refuse to do so if it was the only such event requiring them to have health checks.
That could, in turn, have discouraged designers from showing their collections in the British capital.
The independent Model Health Inquiry -- composed of model managers, designers and supermodels -- was chaired by Baroness Denise Kingsmill.
Also sitting on the panel were Sarah Doukas, founder of Storm Model Management, British model Erin O'Connor and designers Betty Jackson and Giles Deacon.