Yes courses can be accredited by several bodies (Civ Eng at Newcastle has ICE, IStructE and IHT) and you can become chartered and even a fellow of several institutions. It is standard though to just aim for one institutions chartership especially in your twenties.
There are no best ones to be accredited by, you usually get accredited by the institute that reflects what you graduated with and the company you work for.
Graduates of Civil Engineering would usually go for Civil Engineering although if the specialised in structures they may work with their company to get chartered by the IStructE or if they work in transportation they may aim for the IHT.
I would advise you do not worry much about this. You will learn more about it at university and when you graduate. You will not become chartered until you are at least 4 year past your graduation.
A course that is not accredited may make it more difficult for you when you graduate and want to gain chartership as you will have to prove that you have the capabilities in other ways (mainly work experience, submitting work and taking more exams)
If you want me to answer any other questions then I will try to answer them