The flood-damaged structure was originally due to be opened to the public on Friday, but the cold spell has pushed the date back to next Tuesday.
There is a chance it could be unveiled on Monday if temperatures rise significantly over the weekend.
Contractors have been forced to install heated tents over the bridge to allow work on the parapet wall on the upstream side, as the low temperatures have prevented the mortar from hardening.
The bridge was closed after it collapsed during heavy rains last winter, effectively splitting the town in half. The two-year repair project has been condensed into just over 12 months and the Grade II-listed structure, which was build in the 18th century, has been widened during the work.
Coun Don Mackenzie from North Yorkshire County Council said:
“We know how important it is for the town to get this bridge open and we have done everything possible to complete the project - which of its kind would normally take about two years - in half the time. We believe from current discussions we can open the bridge on Tuesday but this complex and challenging process becomes even more difficult in very cold weather.
"We will, of course, keep everybody updated about progress. There is not much longer to wait and we must ask the people of Tadcaster, who have shown such patience and fortitude over this past year, to bear with us for just a short while longer.”