This is a tricky one - I feel for you. But it's excellent that he has a mother who wants to address this. The trick would be to get him doing it when he's seven or eight, but that is not appropriate and in the context looks facetious.
As you're in Cumbria, I'd suggest starting on country lanes and building up from there. Cumbria must be one of these islands' cycling paradises and there must be plenty of locations where the pavement is not an option.
Ours are now nineteen, seventeen and fourteen and apart from the one who wants a 'training partner' (someone to humiliate) they are not all that keen on riding with a parent. To them, my presence strikes the image of a patrician old codger with a pipe and plus-fours. So riding with a parent (and being told how and where to ride by that parent) might not seem 'cool'. Even when I do ride with them, I keep my gob shut these days.
I imagine that at nineteen he will do what he does regardless of your suggestion - and even if he conforms in your presence, he won't when alone or with friends.
Setting an example (as it seems you do) and treating the road as a normal place for bicycles (as you do) is an excellent start. Getting him (bribing him?) to do errands for you by bike might help. I imagine you've been cycling all his life (I may be wrong) so unless you've been up-ended by an errant artic in his sight, he ought to see the road as the normal habitat for a bicycle. Get lots of positive references to road riding into chats.
Does he drive? Mine were all cycling on fairly horrid and scary roads far too young, but two of them now drive and it has altered their perspective about what is safe and what is not. Both the drivers are now more confident cycling on the road than they were before they got behind the wheel. Just a thought.
This is where I might offend the faithful.... You say he gets nervous about your use of primary. I am not a big advocate of its use other than in very particular circumstances. I don't encourage my children to use it (all ride in London and the country). I see riders who I think use it without consideration for other road users or their own safety, which is not to say it is always bad. I can well imagine that there are times when a casual cyclist might see it as either unhelpfully holding up motor vehicles or just ramping up the risk. Not that either of those is the case, but that it might appear that way. If you really want to get him out on the roads and he freaks at your use of primary, would it be worth compromising on that when you're out together? Sorry if that offends.
Ultimately, it might be like having a nineteen-year-old who smokes. They're too old for you to forbid it and they won't listen anyway... so just keep reinforcing the positive messages and be very, very, very patient. He will work it out, but in his own time.
Until then, (although it is a pet hate of mine too) is it really so bad? He is out and about on a bicycle. That is already significantly preferable to the alternative.
Finally... One of mine (the one who races bikes) has mild dyspraxia. One of his 'dislikes' is unpredictability. On one or two rides in North London in heavy traffic he's been really uncomfortable with how close cars came to him and seemed to swoop around him. If your son really doesn't like riding in traffic, might this be some of the cause. My boy with dyspraxia will hurtle down hills at 38mph, but would rather not pootle in heavy traffic alongside three lanes of cars.