I am a UK resident and own shares in a company listed on the NSE of India - does anyone know a broker I can use to sell some of these shares?

Are you a foreign citizen? If yes, it would help if you could tell how you got possession of these shares. Did you buy them yourself? Also are you holding physical shares or do you hold them in dematerialized form?

If you’re an Indian whose residential status has changed, you can open an NRI account at Zerodha. Here’s the process to get started: http://zerodha.com/z-connect/zerodha/open-an-account/nri-trading-on-indian-stock-exchanges

In case you are holding shares in physical format, you will have to dematerialize it before you can sell it through a broker online.

Are you holding the share certificate or is it in your demat account ?