Local MP Fleur Anderson says the seven-year closure is causing congestion in the wider Putney area.
The Port of London Authority said the clean up project was "pretty gross" but well worth it.
Two men were arrested on suspicion of arson after the bush fire at St Paul's School, the Met says.
The west London bridge has been closed to vehicles since April 2019, when cracks appeared in its pedestals.
The idea was among six put forward to resolve the structure's poor state of repair.
MP Fleur Anderson said issues in the five years since the bridge closed may further delay reopening.
Redbridge Council says there is no money to fix Broadmead Road Bridge in north-east London.
A boat carrying West Ham fans crashed into the bridge adding to the mounting costs of its restoration.
The 137-year-old bridge remains closed to motor vehicles but a cycle lane will open for 10 weeks.
