Student visas cannot be extended beyond five years for studying at a degree level (Bachelor's or Master's degree). There are some exceptions and it can be extended if you are accepted on a PhD or MPhil course. On successful completion of studies at a degree level you can apply for a Graduate visa which will allow you to take employment in the UK. Similar to the student visa, the Graduate visa does not lead to settlement, unless your total continuous period of residence reaches 10 years. However, the Graduate route will give you a better chance to find sponsored employment and potentially switch into the Skilled Worker route which will lead to settlement after five years, or when you reach the total 10 years of residence – whatever happens sooner.
If you studied outside the UK but graduated in the last five years from one of the Universities listed on the Global Universities list for the year you graduated, you may qualify for the High Potential Individual visa which will allow you to work in the UK for two or three years (depending on your degree level). This visa does not lead to settlement but gives you a chance to switch into sponsored employment or another route leading to settlement.
Applications for a UK visa are normally made online and supporting documents are submitted to the Home Office in electronic format. Whether you are applying from abroad of from within the UK, you will be required to submit your biometrics. For this you will have to attend an appointment at the visa application centre or at a designated centre in the UK. The standard processing time for the visa varies depending on the type of application. In some cases it is possible to expedite the process by paying an extra fee.