If you are an EU national or you are from Switzerland, you are able to work with no restrictions during your studies. You do not need to apply for special permission to work. Croatian nationals no longer require a registration certificate to work in the UK, as of 1 July 2018.
Do note that if you were previously required to obtain an accession worker authorisation document (as a Romanian, Bulgarian or Croatian national) under the 'work authorisation scheme', it is important that you keep the document (s) you were issued safe, in case they are ever relevant in the future.
Non-EEA International Students
Tier 4 visa holders who are studying full time at degree level and above are likely to able to work 20 hours during term time (please refer to King's academic calendar) and full time during vacations. You should read Tier 4 Policy Guidance paragraph 'Can I work whilst in the UK' for details on how many hours you are allowed to work and the type of work you are allowed to do. The immigration rules relating to work are very strict and breaking them, even accidentally, can result in your visa being cancelled and your removal from the UK, so you should ensure you adhere to the rules.
Please understand for Tier 4 holders who are studying a master programme at King's, summer period is for your dissertation and therefore it is still deemed as term time. King's do not give exemption during this period to Tier 4 students, which means Tier 4 students need to wait until the end date stated on their CAS to take any full time work.
If you hold a Short Term Study or Tier 4 part time visa, you are not allowed to work anytime whilst you study in the UK.
If you hold a different type of visa you should check the working conditions on that visa carefully and seek advice if you are unsure as to what work you are allowed to do. UKCISA guidance is a good source to check out the rules and if you have any queries you can contact the Visa & International Student Advice Team.
You may find it useful to have a look at our Advice Guide for international students working during your studies.