Not so. Being on the electoral register of a property is absolutely not a requirement to get a mortgage.
Electoral roll information is used two-fold in mortgage applications. Firstly being on it enhances your credit score, as it brings back more information on you from the credit reference agencies, building up a better picture of your history. Lenders like this. Secondly it can act as an alternative to providing proof of residency in the form of a utility bill. Many lenders will not require address verifivation if you are on the electoral register at the property from which you are making the application.
You may experience difficulties with regards to the late payments, however if they are less than 1 months late, they should not show as arrears on your credit file and should not be anything to worry about.
Banking with Nationwide gives you the best chance, provided you have run your affairs with them well. As well as your credit score and application details, you will have a "behavioural score" with them internally to show how good a customer you have been. If you have run your account well and not gone over your overdraft limits, etc, you will have a better chance than someone else in the same position as you who banks elsewhere.