The council only learned of the plans through hearsay earlier this month. British Nuclear Fuels Ltd had informed the wrong council, Barnet's neighbour Brent LBC.
BNFL is not obliged to consult local officials and is only required to inform the council because it is the emergency planning authority.
The used fuel will be transported in steel containers from power stations in the south-east and brought together at Cricklewood. The containers will then be taken to Sellafield.
The council has demanded BNFL change its plans.
Liberal Democrat group leader Jack Cohen said: 'We are trying to get across that Cricklewood is not the right place for nuclear waste to be stored. If it's got to be stored somewhere they could pick an area which is not close to businesses and residential properties.'
Barnet is also angry at BNFL's failure to give it details of the operation.
Mr Cohen said: 'They are pleading confidential information about routes and security. All they're prepared to say is that it's an entirely safe operation.
'They've given out as little information as possible. Their idea of consultation was two advertisements in July written in officialese, in the smallest possible type, with no mention of nuclear fuel.'
BNFL said the fuel would be stored for only a few hours at a time and 'used nuclear fuel has been transported through London in complete safety since 1962'.
A spokesman said Cricklewood was 'very well strategically placed' for the operation, adding that BNFL was still collecting the information required by the council.