Here's a list that will upset almost everyone:
Avoid any of the twins bigger than 500cc. Go for a unit engine if you can - it's the one that does not have a separate gearbox. The 500 is a nice, sprightly bike and will give you some idea of why Triumphs were so popular. The bigger twins are fragile and vibrate.
The other one to watch is the Tiger Cub. This is a nice little single that doesn't outperform its brakes (too much). Run away from anything that claims to be a Bonneville or a Trident. They are money pits.
Made a range of pretty good singles, particularly the unit ones. Some of the twins are good, but suffer from popularity. BSA were masters of modular construction, so the same parts often turn up on several different models. Gold Stars and anything with Rocket in the name is usually overpriced and explosive.
Had the name, but not the ability. Norton are the company that tried so hard, struggled so much to innovate, and never made it. Most models are mechanically strange and unreliable. Anything with a Featherbed frame was usually cannibalised to make a better bike.
Their singles used a strange clutch and couldn't keep their oil inside the primary chaincase. That aside they are mostly nice bikes, but they suffer from over-pricing. The LE is a bizarre alternative for anyone wanting something very different.
MATCHLESS AND AJS
They were usually interchangeable badges on the same bike. They made some nice singles and some fragile twins. A decent 500 single is probably the archetype of a British thumper.
Everyone knows the Bullet, which is now made in India. They also made some brutish and fragile twins, and some very nice small singles. The Continental was actually a decent bike, but would have been thrashed to death before falling into your hands.
Made some very nice and sedate singles. They also made the Arrow, a two-stroke twin. Despite its odd looks, its a pretty good bike and could so nearly have taken-on the Japanese, if only yadda yadda yadda...
The Big Pussy from Cleckheaton. Strange engineering, ridden by crazed enthusiasts. You have to love them to want one. Came in 600 and 650cc guise. The only difference is that the 650 burns even more oil than the 600.
Not a make of bike, but a maker of engines. If you want something that will be cheap and practical, find an ancient Brit with a Villiers engine. You may hate two-strokes, but most Villiers units are tough and reliable. With the money you save on buying and running the bike, you can go to a lot of places. Villiers sold engines to every manufacturer who didn't make their own, and a good few more besides, so they account for almost all of the rest of the British bike industry.
Specialist country. If you want one of these, it's because you used to have one or you lusted after one. You're on your own!