Student UK visa after 3 years of study visa expires. ?

Absolutely NO way to remain & work. You are REQUIRED to return to your country of citizenship immediately upon completion or termination of full-time studies. Post-Study Work visas were abolished years ago. And there are NO listed shortage occupations eligible for employment visas for new/recent grads, even those holding advanced degrees. You must have years of post-degree experience PLUS have attained significant promotions to managerial/executive level positions to have any chance of qualifying to work in UK.

You will need a STEM degree in a role that the U.K. Employer cannot fill from U.K. residents.

A visa does not cover any type of work.

A student visa is there to enable you to study f/t, most allow up to 20 hours p't work to enable you to help support your living costs.......... it really depends on what degree you study, what grades you get and if you are offered a full time position in a field of work where there is a short supply of candidates whether you could remain and work on a PSW visa which is 2 yrs

There are no post-study work visas for the UK.

If you want to study, you have to apply for a new visa. If you want to work, you have to apply for a new visa. You can find the requirements for all visas on the UK government website.