Depending on your role, you'll need SC or DV clearance I think. The nuclear industry might have a different set of clearances to do with working on List N sites, but the principle behind getting them will be the same as SC/DV.
SC requires your employment and accommodation history for 5 years of you and your partner. Your parents address, some financial information and that's about all. They'll check its all valid and do some criminal checks and check your not affiliated with extremist groups like the BNP or ISIS or whatever. As long as you're not a criminal, and not so far in debt that you'll be a target for blackmail/coercion, you should get SC.
DV requires all of that but with stricter standards, drug tests, and interviews with your family and I think some close friends. I don't think they check grandparents.
If you get it or not depends on the industry you go into. For example my partner is Chinese, and so I wouldn't be able to get DV clearance for anything remotely related to China, but I would be able to do so for other things.
They're also willing to accommodate certain things. I know of people who admitted to using cocaine in the past, but got DV clearance on the promise that they won't use again (and a clear drug test). However one of those people had their clearance stripped on renewal as this promise wasn't kept.
In short, the majority of normal folk can get whatever clearance they go for.
Source: Used to work for a defense companies.
