1. The first thing you need to do is to apply for a Permanent Residence Card. You will need one for 12 months before you make an application for British citizenship and a passport. In order to definitely stay in the UK after Britain leaves the UK you will need a passport. However, you need to be lawfully resident for five years, not including time spent as a student, in order to apply for British citizenship.
2. So, the two years you spent studying in a college, if you were a full time student, will not count towards the five years lawful residence. So, if you were studying full time for two years, you now only have been four years in the UK which will count towards getting citizenship. However, if you get a permanent residence card, you will need 12 months on it before you apply for citizenship.
3. Here is a link to applying for a permanent residence card on the UK Government website https://www.gov.uk/apply-for-a-uk-residence-card/permanent-residence-card. Be aware additionally, that you will have to have been paying National Insurance for the five years you were resident here, or else you will have to have had private health insurance. Be aware that your two years as a student can count towards the necessary five years, if you had private health insurance during this time period. So, this is an important point.
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5. I have never heard of any principle that if someone pays tax it includes someone else. The issue is not tax, but National Insurance contributions. Your Mum paying National Insurance contributions will never include you. Secondly, the Visas & Immigration department never allow someone to purchase health insurance cover retrospectively. So, this is not an option. I know of no other way you can get this health cover if you didn't have it at the time.
6. You now need to apply for a Permanent Residence Card. You will hold this for one year and then apply for citizenship in one year's time, after you have had the Permanent Residence card for one year.