What will show up on the credit check is exactly the same as what you can see yourself if you pay a small sum to a company such as Experian to see what is on your credit file. It might be worth doing this initially as the fee that the letting agent charges will be a lot more than what Experian charge and it will either put your mind at rest or save you paying money over to the letting agent if it looks poor.
As a landlord, if the defaults on payments are in the past and have been paid off since I wouldn't worry about it. If they are still owing then I'd probably ask you to find a guarantor. This would have to be a house owner who would pass the credit check.
CCJ's are probably a no-no as far as renting is concerned - unless the landlord is desperate!