The process for applying for a post-study work visa in the UK is as follows:

1. Choose the visa category: The first step is to determine which type of visa you are eligible for. If you have recently completed a course of study in the UK, you may be eligible for the Tier 1 (Graduate Entrepreneur) visa or the Tier 2 (General) visa.
2. Check eligibility: Make sure you meet the eligibility criteria for the visa category you have chosen. This includes requirements such as having a valid Certificate of Sponsorship, having enough funds to support yourself, and meeting the English language requirements.
3. Prepare documentation: Collect all the necessary documentation, including your passport, qualifications, financial evidence, and English language test results.
4. Apply online: Submit your application online through the UK government's visa and immigration website. You will need to pay the visa fee and provide your biometric information, such as fingerprints and a digital photo.
5. Attend a biometric appointment: You will need to attend a biometric appointment to have your fingerprints and digital photo taken. This will usually be at a UK Visa and Citizenship Application Services (UKVCAS) centre.
6. Wait for a decision: The UK visa processing time can vary, but you should receive a decision within three weeks of your biometric appointment. If your application is successful, you will receive a visa that allows you to stay and work in the UK for a specified period of time.
7. Start work: Once you have your visa, you can start work in the UK in the role specified in your Certificate of Sponsorship. You must also keep your visa up-to-date and comply with any conditions attached to it.

It is important to note that the post-study work visa requirements and application process may change from time to time, so it is always a good idea to check the latest information on the UK government's visa and immigration website before applying.