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 indeed 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 (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 could therefore apply for UK entry clearance (a visa) in your home country before travelling to the Republic of Ireland.
If you do enter the UK from the Republic of Ireland from 11pm on 31 December 2020 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 for six months (if you are not an excluded person, as outlined in Article 3 of the Order, as amended) 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, as amended, 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 74 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 S2 Healthcare provisions (see pages 60-61 of the Home Office guidance on the Common Travel Area) or under provisions for permitted page engagements (see page 62 of the Home Office guidance on the Common Travel Area) or under the creative worker route (see pages 62-64 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: If you entered the UK before 11pm on 31 December 2020, the Immigration (Control of Entry through Republic of Ireland) Order 1972, as amended the period of leave that you will automatically be considered to have (if you are a non-visa national without entry clearance and are not an excluded person, as outlined in Article 3 of the Order, as amended) is three months. This is reduced to 7 days if you entered the Republic of Ireland from the UK at a time when you still had permission to be in the UK but that permission expired while you were in the Republic of Ireland.
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.