My daughter was born in the US but because my wife and I are British we were able to register her birth in the UK and hence she is a dual UK/US citizen. This gives us the advantage that she does not need a Tier 4 Student Visa to study in the US. To be fair I have not researched the Tier 4 Visa process but details can be found on your colleges website. Essentially the college becomes your visa sponsor. York St. John’s web page (link) gives details. If you are applying from outside the UK, the cost of the visa is £335 (plus the Immigration Health Service Charge of £150 per year).
The Health Service Charge is so you have access to UK doctors and Hospitals for a fixed fee. The government has clamped down on “healthcare tourism” where people were traveling to the UK to take advantage of the free socialized healthcare.