Having only been down the A59 from Starbeck to Knaresborough, I wouldn't fancy the ride up in the opposite direction. The (Beryl Burton?) cycleway & lanes to Bilton and the Nidderdale Greenway back were a much more scenic and less hilly option for me. Plus taking in the the extension down to Riply on a second trip later in the day.
I usually use the up hill cycle track, it starts a short way up the climb but as the road quickly widens out to two lane there is rarely any close-ish passing traffic.
The steeper nature of it compared to the road is a bit of a killer and when it's quiet I have stayed on the road but the surface has recently got really rough.
I dislike using the Beryl Burton on the road bike due to, one, having acquired punctures and, two, having had to stop on the steep bit going up hill. I do use it downgrade when on the ebike.
The cycle track next to the A59 ends immediately before a road off to the right which comes to the NCN beyond Beryl Burton.
Alternatively continuing on the A59 to a right turn at a set of traffic lights brings you to a bridge over the Starbeck branch of the Nidderdale Greenway immediately beyond which their is a steep decent down to it.