How to become a visiting student in a UK University
There's no simple formula, and each university will have its own visiting student schemes. This being said, there are usually classic methods for visiting a UK university:

Enrolling in a degree

Obviously you can apply to be enrolled in a degree, pay your tuition fees and study for the degree in the UK. You can usually get a Visa via this mechanism (except for London Metropolitan University which is no longer allowed to issue visas by the UK Border Agency).


Duration: 1 to 4 years.
Cost: a lot. ~£4,000 to £10,000 for EU students, ~£20,000 for non-EU students.
Requirements: apply to target uni. Proof of academic competence required, usually good grades and a BSc/BA for a MSc/MA position.


Applying for an semester abroad as an Erasmus student

You can use the Erasmus Programme to visit a EU university as a EU student. Some non-EU universities located in Europe may also be eligible.


Duration: usually 3 to 6 months.
Cost: free, paid by your university and the EU's Erasmus mobility budget.
Limited scholarships available, usually ~€400 / month
Requirements: must apply to your uni of choice, selection process varies. Your uni often must have a prior agreement with the target uni.


Applying as an exchange student

You can apply outside the Erasmus scheme if you uni has an exchange agreement with the target UK uni.


Duration: usually 3 to 6 months.
Cost: depends on agreement, can be free or can requirement payment of tuition fees.
Requirements: depends on agreement.


Applying as a visiting research student.

You can get opportunities to visit foreign unis as part of your PhD, a bit like an internship. This is fairly common across the world.


Duration: 3 to 12 months.
Cost: free.
Requirements: must be enrolled in a PhD-level course, must have found a supervisor to welcome you in target uni.


Applying for an internship.

You can also join a uni as part of an internship programme, if a position is available in a research group. This normally does not involve any teaching, or in marginal amounts.


Duration: 2 to 6 months.
Cost: free.
Remuneration: depends on uni, not guaranteed... ~£330 per week in some London universities.
Requirements: there must be an internship position available, and you must be the best available candidate. You must work.


Applying for a volunteering position.

You can also join a uni as part of an internship programme, if a position is available in a research group. This normally does not involve any teaching, or in marginal amounts.


Duration: depends.
Cost: free.
Remuneration: none.
Requirements: there must be a volunteer position available, and the supervisor must agree to take you in. No deliverables required.