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In regards to Tottenham Court Road, it would not be at risk of increased traffic levels or gridlock. Camden and TfL are currently undertaking a year-long project to restore two-way traffic to the Gower Street-TCR gyratory by making TCR accessible to buses and bikes only during weekdays and moving all other vehicular traffic to Gower Street:
http://www3.camden.gov.uk/westendproject/
Beyond that, the Central London BIDs, in their response to the Mayor's Transport Strategy, have called for a comprehensive master-planning of the various pedestrianization and traffic-calming projects so as to make sure any displaced traffic can actually go somewhere.
As it pertains to Oxford Street, though, Westminster City Council withdrew their support for full pedestrianization shortly after the local elections and have said they will take the summer to look into other options.