You are eligible to work in the UK, and therefore don’t need to obtain a work permit, if any of the following apply:
You are a British citizen
You are an European Economic Area (EEA) citizen – see list below
You are a Swiss national
EEA countries: Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Liechtenstein, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden and the UK.
Nationals of certain countries, such as Romania, Croatia and Bulgaria, may require special permission to work in the UK.
You will also need to apply for a UK residence card or be able to prove your right to work in the UK as an EU citizen (for example, a valid UK National Insurance number).
For more information related to EU, EEA and Commonwealth citizens and your rights to working in the UK, please visit: https://www.gov.uk/browse/visas-immigration/eu-eea-commonwealth.