This talk of A roads is worrisome . . . Will the route mainly be quite quiet?
yes. Mostly.
First of all - traffic levels outside the southeast of England are generally lower, and, in Scotland, much lower. That doesn't necessarily make them safer, because people in Northern Scotland drive like loons, but, taken as a whole, we go down roads you'd think yourself lucky to be on if you're used to the southeast.
Having said that....there are some stretches that will be less pleasant.
If we go via Bingham and York
a) the A508 from Northampton to Market Harborough has lots of traffic, and is not very wide, but we will aim to get off it by breakfast time on Saturday morning, when most of the cars will be coming south. I'll be at the back, giving the cars the hard stare.
b) we'll clear the A6097/A614 early on Sunday morning and while the A60 and A19 in to York are busy* (*Reduced Northern Scale), but not overfast, and it will be a Sunday. If the traffic is heavy we can take a diversion north of Selby which will add a mile or two to the trip. To give you an idea - (moralists please look away now) I rode down ten miles of the A60 on the phone - there's no way I'd do that in the daytime in the southeast
c) the A19 north of York is busy, but we will be off it very early on Monday morning - again, going against the flow
d) the A167 is not very busy, and we will be on it for a short time
e) the A68 (aka The Hell of the North) is not very busy, and has few commercial vehicles south of the A697 junction, where it becomes a wideish smooth road. It's busy going in to Edinburgh, but we'll start early in the morning, go via Nether Blainslie (I'll be looking for the pothole I hit in 1980 and 2003) and get over the hills before a late breakfast. We do take a diversion at Pathhead, and go in to the city on the A7, which is a bit like the Romford Road, but not as busy.
f) I'll take local advice on the A90 toward the Forth Road bridge, because we'll be on it late morning/early afternoon
g) There are stretches of the A9 between Perth and the Kessock Bridge near Inverness that are difficult to avoid (we'll mostly be travelling on the 'old A9'). There are heavy trucks on this road, and if it rains, or there's a crosswind, it's no fun, particularly over Drumochter - having said that, the last time I went up it the southwesterly was so strong that I barely had to pedal.
f) Snorri is going to disagree with me here, but I reckon the A9 north of Tain is kids' stuff. Put it this way - There's No Way It's Essex
My plan is to publish the route with projected times, and people can judge for themselves - but I do know the entire route, and there are many, many times when you wonder if you're in the same country.