This information is for nationals of the EU, EEA, 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 to study.
As a non-visa national, you can normally travel to the UK and apply when you arrive for immigration permission to enter as a 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 (including 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 could therefore apply for UK entry clearance (a visa) in your home country before travelling to the Republic of Ireland.
If you enter the UK from the Republic of Ireland, and you are a non-visa national, you will in most cases automatically have permission to be in the UK for six months (if you are not an excluded person, as outlined in Article 3 of the Order, including any amendments that have been made to the Order.
This is reduced to two months 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, including any amendments that have been made to the Order, and further amended by the Immigration (Citizens’ Rights etc.) (EU Exit) Regulations 2020. Page 57 of the Home Office guidance on the Common Travel Area) lists groups of people who do not require this automatic permission, as they have permission under other provisions. The Home Office guidance on the Common Travel Area has a flowchart on page 73 which illustrates what type of automatic permission someone will have, and for how long, based on their circumstances.
There are different automatic permissions for those entering the UK under provisions for permitted page engagements under the Visitor route (see page 61 - 62 of the Home Office guidance on the Common Travel Area) .
Page 44-45 of the Home Office guidance on the Common Travel Area lists travel documents that you will be required to produce to confirm your nationality and identity "if you are encountered by an official as part of an intelligence led control on arrival from Ireland". Most students will be using a passport (only a limited group of people can use a national identity card as a travel document - see page 45). If you are relying on your passport, it must be valid for the whole duration of the time you are in the UK.
You must not engage in any 'occupation or employment' except an exempt work activity, for which you may receive a permitted payment. Occupation or employment that is prohibited includes: undertaking employment; doing any work for an organisation or business; establishing or running a business as a self-employed person; undertaking a work placement or internship; undertaking any direct selling to the public; fulfilling a contract to provide goods or service. There is nothing preventing study.
You will not be able to apply for permission to stay beyond the six months (of automatic permission) from inside the UK as a Student. If you need to be in the UK for more than the six months, you will need to leave the UK within the six month period and apply for Student permission, in the usual way.
Note that any immigration permission you are given on arrival in the Republic of Ireland is for the Republic of Ireland, not for the UK.