Stay back:
Graduate Route (Post-Study Work Visa):
Duration: The Graduate Route allows international students to stay in the UK for 2 years after completing their studies (or 3 years if they have completed a doctoral degree).
Work Permission: During the Graduate Route period, international students can work, seek employment, or start their own business without the need for employer sponsorship. There are no restrictions on the type of work, and students are not required to have a job offer to apply for this visa.
Switching Visas: Graduates on the Graduate Route can switch to other work visas (like the Skilled Worker Visa) from within the UK if they find a job that meets the specific visa requirements.
Dependents: If you’re on the Graduate Route, your dependents (spouse, partner, and children under 18) can join you in the UK and work or study without restrictions.

PR (permanent Residency)

To get permanent residency (PR) in the UK after study, you must first have a valid UK visa and must fulfil the eligibility requirements. These include:
You must have spent five years residing and working in the UK.
You must be 18 years or above.
You must have a good character and no criminal record.
You must be able to support yourself and your dependents financially without recourse to public funds.
You must pass the Life in the UK test and a recognized English language test.

Once you have completed five years of residency in the UK, you can apply for ILR.

Here are some tips for increasing your chances of getting PR in the UK after study:
Choose a course that is in high demand in the UK job market.
Gain work experience while you are studying.
Network with people in your field.
Apply for jobs as soon as you graduate.
Take the Life in the UK test and a recognized English language test as soon as possible.