If I come here to study a Masters course, but have completed my undergraduate studies in a different EU country, will my degree still be recognised?

The UK, along with 47 other European countries including many outside of the EU, is part of the Bologna Process which agrees on the comparability in the standards and quality of higher-education qualifications. This process is not part of the EU (although includes all EU countries) and there has been no suggestion that the UK will leave this mechanism.