A Start-up visa is designed for entrepreneurs who want to start their business in the UK. It is issued only for two years and thus does not lead to granting citizenship. Your business project is expected to be developing so when your Start-up visa expires, you will be able to switch to an Innovator visa easily, provided your project is successful.
Student visas do not guarantee citizenship either but they give you a chance to add the years spent in the UK with these visas to overall years of residence in Great Britain – 10 years are necessary to obtain ILR. Note that you should not leave the UK for a long time after you have completed your studies. In other words, you need to find a company to do your internship while studying in the UK in order to apply for a work visa and stay in Great Britain.
Takeaway: as a beginning entrepreneur or a student you will hardly be naturalised but time spent in the UK with such visas can be included in the 10-year period necessary to apply for ILR if you have not left Great Britain for a long period of time.