This information is for nationals of the USA, Japan, South Korea and other non-visa nationals who are planning to travel to the UK via the Republic of Ireland, and enter the UK as a short-term student.
As a non-visa national, you can indeed normally travel to the UK and apply when you arrive for immigration permission to enter as a short-term student (or other type of visitor) for up to six months. Non-visa national students coming to the UK for a short course often choose to do this rather than applying for entry clearance before travelling. However, if you arrive in the UK from elsewhere in the CTA (this includes the Republic of Ireland), you will not have the opportunity to do this because there is no immigration control at your UK arrival point. You should therefore apply for UK entry clearance (a visa) in your home country before travelling to the Republic of Ireland.
If, despite the above advice, you do enter the UK from the Republic of Ireland with no specific UK immigration permission, and you are a non-visa national, you will in most cases automatically have permission to be in the UK only for three months, not for six months. This is reduced to seven days if you entered the Republic of Ireland from the UK at a time when you still had permission to be in the UK and that permission has since expired. This provision was established with the Immigration (Control of Entry through Republic of Ireland) Order 1972, as amended in 2014.
The three months' permission allows you to study, but it may not be enough time to complete your short course, but you cannot extend your stay in the UK as a short-term student or Tier 4 student. If you need to be in the UK for more than the three months, you will need to leave the UK within the three month period, then re-enter asking for immigration permission to enter as a short-term student, for a maximum visit of six months.
Note also that any immigration permission you are given on arrival in the Republic of Ireland is for the Republic of Ireland, not for the UK.