The improvement works at Junction 47 will run until 26 March and are currently more than £2.3million over budget and five months over schedule.
North Yorkshire County Council said the project has hit a "series of unforeseen delays" as it also warned motorists that some overnight road closures will now be required in order to complete the "finishing touches".
Councillor Don Mackenzie, executive member for access at the county council, said: "The finish line is in sight for this important scheme which will support the future growth of the Harrogate and Knaresborough areas.
"The scheme promises to reduce congestion and improve road safety at this major junction. It is a great example of how we are delivering on our aim to improve east-west connections across North Yorkshire."
As of September 2021, the project had cost around £10million compared to an original budget of £7.7million.
The county council said the latest estimated costs are subject to "detailed discussions" with its contractor and will be made public once an agreement is reached.
Completion of the project was originally scheduled for October 2021, but "poor" ground conditions and the Covid pandemic have been blamed for the delays.
Construction workers also found great crested newts on the site which caused further delays as the protected species had to be relocated.
The project has involved the widening of three of the four slip roads onto and off the roundabout to increase capacity, as well as a number of junction improvements including traffic signals.
To the west of the junction, a lane has also been added for traffic travelling east so there will be two lanes in each direction.
Final repair works on the A168 bridge along with the resurfacing of the A59 will run from March 14 to 26 under overnight weekday closures.
Once the traffic signals are installed, there will be a period where the signal timings are adjusted by engineers. The county council said this may result in some extra delays during March and April.
The project has been supported by £2.47million from the government’s Local Growth Fund, secured by the York and North Yorkshire Local Enterprise Partnership.
There have also been cash contributions from the county council, National Highways and developer Forward Investment LLP.