I start a lot of these traffic answers by saying "I am not a traffic officer, it's not my speciality, I stand to be corrected".
That said... The road is hedgerow to hedgerow. Not curb to curb. So traffic travelling along the main road has priority over those who wish to turn off it into a minor road, or come out from that minor road.
If you are cycling legally on a cycle path then you should under that frame of reference have priority - however the number of drivers who would realise, accept or acknowledge this is likely to be very low.
The issue would be further complicated by the cycle paths that have those "give way" markings before every junction. Logic (which doesn't always necessarily apply well to the costs) would suggest these give way markings mean two things :
1. Clearly my earlier assumption of priority is true - otherwise why would these markings be needed at all?
2. When they exist, the cyclist no longer has priority and should give way.
I would imagine if taken to the nth degree or would need some sort of stated case to give a definitive answer, and it's one we are unlikely to get.
Personally... I would use the road precisely because I don't wish to keep stopping and giving way, because regardless of whether I 'need' to our not, I would give way at these junctions for my own safety.
That's my take on it anyway.