The road bridge over the River Wharfe at Tadcaster, near York, partially collapsed in December.
Currently the river can only be crossed using a temporary footbridge.
North Yorkshire County Council said "persistent and unusually high river levels" had delayed the work, which had been expected to finish by next month.
In a statement, the authority said: "The County Council has learned in the last few days that the primary cause is damming of the river by Network Rail five miles downstream at Ulleskelf for work on a railway bridge.
"Although the Environment Agency issued a permit for this work to be carried out, North Yorkshire was not notified about the operation."
Councillor Chris Metcalfe said: "We are extremely frustrated by this development."
Network Rail said it had "agreed the terms of our work with the Environment Agency prior to accessing the river" and was first made aware of the issue on Wednesday.
"After speaking with the Environment Agency today, we have agreed to remove our worksite from the river to allow water levels to drop and essential work to take place at Tadcaster."
The Environment Agency said: "We regulate work in rivers to ensure that any works carried out do not increase flood risk. When we issue a permit we take into account flood modelling data and determine if consent can be granted under Environmental Permitting Regulations. Our decision does not take into account whether one project is more or less important than another, whether they are related to rail infrastructure, road infrastructure, or other purposes."
The government pledged £3m for repair work within days of the bridge collapse, and £1.4m was given by the region's Local Enterprise Partnership to widen and strengthen the bridge.
The completion of repair work to a Grade II listed bridge damaged by winter flooding last year has been delayed until January.