City Clean Air Zones delayed
Vehicle checking software required to make the zones operational and enforceable was expected to be delivered by the government by October.
The cities' councils say the tool has been delayed until December, which will put back the January start date.
The government has been contacted for a response.
Leeds City Council and Birmingham City Council were asked to tackle air pollution as a priority after roads in the cities were identified as likely to fail legal air quality levels by 2020.
Birmingham said its new zone would now not come into force in July 2020, at the earliest.
A revised date for the scheme to start in Leeds has not yet been announced.
In a joint statement, the councils said they "had been on track to implement Clean Air Zones on the basis that a vehicle checker tool, which is being delivered by the government's Joint Air Quality Unit (JAQU), would be ready by October 2019 as planned.
"However, JAQU has now confirmed that the vehicle checker will not be available until at least December 2019 - leaving just weeks before the zones were due to come into force in January 2020."
Leeds City Council said it had successfully met "a number of challenging deadlines" only for the government to "fail to meet its own commitments".
Councillor James Lewis, deputy leader, said: "The government now needs to outline new timescales that they are confident can be delivered in order to give residents and businesses across the country clarity and certainty."
It said it would still be installing the camera infrastructure required for the zone in the coming weeks.
Birmingham City Council's cabinet member for transport and environment, Waseem Zaffar, said sticking with the January start "would be completely unfair on residents, businesses and visitors to the city".
"This is simply unacceptable," he added.