Frankly, this is the weak spot in the Hadrian's cycle route. Once you get to Bardon Mill, you have 2 alternatives, follow the official route 72 uphill, past Vindolanda (the roman fort) followed by a long and pleasant descent into Hexham, or ride along the A69, main road, until Haydon bridge. At Haydon Bridge, go over the railway, turn right and follow the road to Fourstones where you rejoin the 72.
The A69 is not a pleasant road to cycle on, it is quite busy. Personally, I would avoid it but it's up to you.