Halsbury's Laws of England (volume 57 5th edition 2012) Immigration and Asylum Immigration and Asylum on Lexis + Australia is a good starting point for your research into UK refugee law. Halsbury's provides up to date commentary and links to relevant caselaw and legislation.
To get there, scroll to Immigration and Asylum in the alphabetical list, then click the + button which opens the menu - click on 3 - Asylum, Humanitarian Protection etc. You can then browse to the relevant entry eg: Establishing Refugee Status.
Refugee and asylum cases are heard in the following courts and tribunals in the United Kingdom:
Cases can also be heard in the European Court of Human Rights.