The Pennines - border area between England and Scotland - are beautiful, and easy to reach from Newcastle (North Shields), so that's an area I can definitely recommend. It's fairly empty, not many cars, and I've always found drivers in the UK very courteous.
You'll find many single track roads in the countryside. Cars will have to stay behind you if they haven't got a good view ahead of them to safely overtake you. In my experience drivers have always been very patient. You'll have to accept being on the same road as cars, as separate bicycle lanes are nearly non-existent. But then there are so few cars, the situation can't be compared to that in The Netherlands.
What I would recommend for Britain is getting a rear view mirror. Not many cyclists in flat Netherlands use one, but especially going downhill at high speed, when you want to keep your eyes on the road in front of you, you'll still want to be aware of traffic coming from behind. And of course your brakes need to be good in the hills!
Cycling with full camping gear is really tough in the hills. Whenever I saw fully packed cyclists in Scotland, having an arduous time, I tried to make out their brand of bicycle. It was usually a Dutch brand, so I knew; 'silly Dutch people'. The scenic roads you'll want to be cycling are never flat, the bigger roads will be, but they're busy roads, and not the reason you came over.
So I would try to organise your stay in a way that would allow for day trips with your bike, unpacked, instead of planning it so that you would be fully packed most of the time, because that would be really tough.